Is 1 = 0.999... ? Really?

[obsrvr524]

@Ecmandu

Obsrvr,

The only reason motion happens in the cosmos is because of infinity: infinity = motion (because it never ends - by definition)

You’re the person here who looks at dictionaries and can’t actually think to this regard, not me.

Just think about it.

Even if true, I don’t think that changes what I said. Infinity is not an action or instruction or function - “Get the bloody words straight”.

[Certainly_real]

@obsrvr524

There is no connector. They are merely side by side portions - like the right and left sides of a piece of paper. How one infinitesimal is distinguished from another is merely by which portion you are focused on or talking about. The distinction is an imaginary line drawn between to portions of an otherwise continuous substance - the North portion of the Sea or the South portion of the Sea.

If there is no connector and there is no separator, then how can the following not be the case: Nothing separates/connects one infinitesimal from/to another.

Every portion (an item in our discussing of the whole of the substance) can have that measure (assuming infinitesimal portions). I don’t know where you get that there can be only one. How many infinitesimal portions are there of a pint of beer?

If I say there is an infinity of infinitesimals in one finite pint of beer, then that either implies:

A) You can have a quantity that is beyond Infinity because other things exist and this would imply Infinity + other things (contradiction)
B) An infinite number of infinitesimals encompass all things. As in every existing thing consists of an infinity of infinitesimals.

A is clearly contradictory. Whether B is contradictory or not is not crystal clear to me.

B suggests that an Infinity of infinitesimals exist. When no existing thing separates/connects one infinitesimal from another, then perhaps B is contradictory because if x is absolutely not separated from x by anything other than x, then can we meaningfully say there is more than one x? Even if we choose to hold on to the belief of there being an infinity of xs/infinitesimals (which is my current belief), I think the following still holds true:

For any given item (finite or otherwise), you either say there are an Infinity of infinitesimals/xs there, or you say the Infinite is there. You cannot say there are 5 xs/infinitesimals here and 6 there. Nor can you say there are 5Inf infinitesimals/xs here and 6Inf infinitesimals/xs there. Can you divide or multiply the quantity of Infinity by anything? I don’t think so. Even if you say you could, it would go something like this. 1/2 of Infinity = Infinity. 2 times Infinity = Infinity. 1/Infinity of Infinity = Infinity. Infinity times Infinity = Infinity. Square root of Infinity = Infinity. Infinity to the power of Infinity = Infinity. Infinity is always the same Infinity.

An infinity of xs can sustain/accommodate any finite number of ys. There is no end to to how big this finite number of ys could be, but this number must be a finite number as it cannot be Infinity.

So, an Infinity of xs cannot accommodate an Infinite number of ys. An Infinite number of xs don’t just sustain/accommodate an Infinite number of xs. They are an infinity of xs. Or as I would say, x IS Infinity. Everything is in Infinity and Infinity is in everything. Only x denotes the measure of Infinite/Infinitesimal. Only x is Infinite in quantity. Nothing else is like this. Given how I have define x and y, to have an endless number of finite numbers (ys), is not the same as being an Infinity (xs). The endless number of xs necessarily = Infinity. The endless number of ys necessarily = not Infinity.

[obsrvr524]

@Magnus Anderson

@obsrvr524

I don’t know why you are not starting with 1 or 0. Why write it backwards (from n down to 1)? Or is that just part of a new age thing in America (I had some pretty old professors)? And I don’t know why you stopped at “n-1”.

Does it matter how you write it down? You can do it any way you want. There isn’t much of a difference between your approach and my approach.

Wikipedians do it the same way that I do except they start with (0).

And the reason we do it this way is because it’s more elegant.

Consider this (your approach):

not -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)
but -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^{n-1})

And now consider this (Wikipedia approach):

(d_n \dotso d_2d_1d_0 = d_n \times 10^n + \cdots + d_2 \times 10^2 + d_1 \times 10^1 + d_0 \times 10^0)

Which of the two expressions look more elegent to you?

So it really is a new age thing - the over concern for appearing “elegant” - a class struggle issue - everything about appearances and ordained truth narratives - focus on the symbols not the reality so we can keep the peasants confused. Gauging by this board - they have certainly succeeded.

@Magnus Anderson

And this isn’t the only reason I write it that way. There’s another one.

As for why I start with (1) rather than (0), there’s a reason for that too.

How can an index greater than every integer be less than (\infty)?

Right. If (\infty) stands for “any number greater than every integer” then a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would be an obvious contradiction.

Not exactly.
You say “any number greater than every integer” - and that alludes to the possibility of more than one number greater than every integer. And that’s ok (sort of).

But then you say "a number cannot be both greater than every integer and at the same time less than any number greater than every integer. That would require that there be only one “greater than every integer” - else a number could be greater than every integer while another number is greater than that one.

You seem to be talking about greater and lessor infinities (why - I have no idea).

@Magnus Anderson

That’s why, as you already know, we have to do one of the following things:

1. we have to say that, in our case, neither (\infty) nor (\dotso) indicate “any number greater than every integer”, but rather, that both indicate one and the same specific number greater than every integer

2. we have to come up with a neutral symbol, such as for example (x), which we can say represents some specific number greater than every integer; and we also have to say that the number of digits in (999\dotso) is (x) (to make it clear, we can use something like (999\dotso_x))

Maybe that is the seed of disagreement. Your (1) doesn’t seem to be true.

Although the symbol (\infty) can possibly indicate ANY infinite “number” (higher cardinality), the “…” ONLY refers to the entire natural number set. I don’t think anyone every uses “…” to refer to a higher cardinality.

And that makes no need for (2).

@Magnus Anderson

My claim is that (999\dotso_x) stands for (9 \times 10^{x-1} + 9 \times 10^{x-2} + 9 \times 10^{x-3} + \cdots) where (x) stands for some specific number greater than every integer.

That doesn’t seem “elegant” to me. It seems a little messy and unnecessarily complicated because of all of the "x - n"s that could have been avoided.

@Magnus Anderson

I also claim that

1. YOUR expression and MY expression represent two different numbers,
2. my expression represents a larger number,
3. my expression represents (999\dotso_x), and
4. your expression represents (\dotso999_x).

That we disagree on. Again apparently you are focused far too much on the positioning of the symbols (much like Certainly true) - ignoring the relevant meaning of those symbols. the “…” cannot reasonably be placed before the string of digits that define what is to be repeated. And even if it is written that way, it still represents the same value that we both have been discussing. There is no difference in the values.

The “…” isn’t about how to construct the symbol or representation. It is about representing the value.

@Magnus Anderson

Now, let me take the following statement of mine . . .

@Magnus Anderson

The non-zero digits in (999\dotso) are digits whose index is less than (\infty) but greater than every integer.

. . . and reword it, like so:

The non-zero digits in (999\dotso_x) are digits whose index is less than (x) but greater than every integer.

But I think I made a mistake here.

Let’s see.

The index of the first digit is (1), so the index of the last digit is equal to the number of digits. Since the number of digits is (x), the index of the last digit is (x). Thus, the non-zero digits in (999\dotso_x) are digits whose index is less than (x + 1) but greater than every integer.

Do you agree so far?

Two issues -

1. Why do you keep saying “non-zero digits” when ALL of the digits are clearly non-zero - “9”

2. If every index is greater than every integer, you certainly have not included either the higher or the lower portion of integers, such as those at “999…” or those down at “…999.0”. It depends on how you set your index.

[Magnus_Anderson]

@obsrvr524

What word would you use to describe this kind of number - 0.999… ?

@Certainly real

Infinity

@wtf

Can you explain your reasoning for this? I think .999… = 1. But even if someone disbelieves that, they are going to say that .999… is just a smidge less than 1. People tend to think it’s “smaller than 1 by an infinitesimal amount.”

1 is finite so a quantity a tiny tiny bit less than 1 is still finite. I don’t follow why you think this is infinite.

If we take it that what Certainly real means by “infinity” is “a number greater than every integer” then (0.\dot9), being smaller than some integers, is not a number greater than every integer, and thus, not “infinity”. (Whether we take it that (0.\dot9) is equal to (1) or less than (1) has no impact on that and thus it’s irrelevant.)

But is it a finite number? I can’t tell because I don’t know what that means. What exactly is a finite number?

I can say that it is not an infinitesimal because the word “infinitesimal” is generally taken to mean “a number greater than (0) but smaller than every number of the form (\frac{1}{n}) where (n) is a natural number”. Since (0.\dot9) is greater than (\frac{1}{2}), it’s not an infinitesimal.

I would say that we have no name for this type of number.

@Certainly real

Are we in agreement that you cannot count to infinity?

My position is that we can. I am, of course, assuming that what you’re asking is “Is it logically possible to count to infinity?” and not “Is it possible to count to infinity in reality?”

[Magnus_Anderson]

@obsrvr524

not -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)
but -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^{n-1})

I did make a mistake but you did not correct it properly.

Let’s correct it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})

The least significant digit (the one that you should multiply by (10^0)) is the rightmost digit i.e. (d_n).

According to you, (256) is equal to (2 \times 10^0 + 5 \times 10^1 + 6 \times 10^2) which is equal to (2 + 50 + 600) which is (652).

So it really is a new age thing - the over concern for appearing “elegant” - a class struggle issue - everything about appearances and ordained truth narratives - focus on the symbols not the reality so we can keep the peasants confused. Gauging by this board - they have certainly succeeded.

If you make things as simple as possible, you are less likely to make a mistake. That’s why elegance (or simplicity) is important.

Why do you think I made the mistake that you tried to correct but failed? (And why do you think you made the mistake that I corrected, namely, that (999\dotso) is the same as (\sum_{i=0}^{i->\infty} 9 \times 10^i)?)

And that’s also why one should not redefine (0.999\dotso) to mean “the limit of the sequence ((0.9, 0.99, 0.999, \dotso))”. Instead of changing the existing meaning of (0.999\dotso), one should come up with something like “lim(0.999…)” to avoid possible conflation.

[obsrvr524]

@Magnus Anderson

@obsrvr524

not -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^0)
but -->> (d_1d_2d_3 \dotso d_n = d_1 \times 10^0 + d_2 \times 10^1 + d_3 \times 10^2 + \cdots + d_n \times 10^{n-1})

I did make a mistake but you did not correct it properly.

Let’s correct it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})
The least significant digit (the one that you should multiply by (10^0)) is the rightmost digit i.e. (d_n).

That is not what I had represented. That is you trying to conflate a picture with a value (the very source of obfuscating reality and subverting the peasants).

@Magnus Anderson

According to you, (256) is equal to (2 \times 10^0 + 5 \times 10^1 + 6 \times 10^2) which is equal to (2 + 50 + 600) which is (652).
Not at all.

You are mapping the first number seen in the picture as “a first number” to make - \2*10^{n=0}). I never said anything like that. And the maths that I took never ever made that kind of conflation.

You are taking what wasn’t wrong and mixing it with your own choice of representation then claiming it to be wrong. That is exactly what is happening in politics - changing the words to make the words of the past to appear bad (“white man BAD!”).

It is the new age socialist/communist thing to do.

@Magnus Anderson

If you make things as simple as possible, you are less likely to make a mistake. That’s why elegance (or simplicity) is important.

But you didn’t. And worse, when you tried - you cast blame on the prior for being “wrong”. That is what they call “virtue signaling” - “Our new way is RIGHT and GOOD and any other way is WRONG! and EVIL!”

@Magnus Anderson

And that’s also why one should not redefine (0.999\dotso) to mean “the limit of the sequence ((0.9, 0.99, 0.999, \dotso))”. Instead of changing the existing meaning of (0.999\dotso), one should come up with something like “lim(0.999…)” to avoid possible conflation.

And that is also a part of the game.

Those already experienced with the old simply shortened it because of widespread understanding. But now in the cancel generation, you proclaim an understanding superior to their obviously flawed rhetoric - even though it wasn’t flawed at all. You merely never learned it but then criticize it.

It is like the Americans claiming that the Brits don’t know how to speak English properly.

Okay. I think I see what is going on here - shifting the meanings of words and symbols to make old things “appear” wrong and new to “appear” right. It is all about appearance - focus on the look, not the substance or reality.

No need to go further with ironing out any actual logic.

It is just ordained NEWSPEAK.

[Magnus_Anderson]

@Magnus Anderson

I did make a mistake but you did not correct it properly.

Let’s correct it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})

The least significant digit (the one that you should multiply by (10^0)) is the rightmost digit i.e. (d_n).

@obsrvr524

That is not what I had represented. That is you trying to conflate a picture with a value (the very source of obfuscating reality and subverting the peasants).

You are supposed to explain what I am doing wrong instead of ranting about politics.

@Magnus Anderson

According to you, (256) is equal to (2 \times 10^0 + 5 \times 10^1 + 6 \times 10^2) which is equal to (2 + 50 + 600) which is (652).

@obsrvr524

Not at all.

You are mapping the first number seen in the picture as “a first number” to make - (2*10^{n=0}). I never said anything like that. And the maths that I took never ever made that kind of conflation.

That’s because we’re indexing the digits from left to right. The index of the leftmost digit is (1), the index of the one next to it is (2) and so on. That’s what (d_1d_2d_3 \dotso d_n) stands for. In the case of (256), (d_1) is (2), (d_2) is (5) and (d_3) is (6).

That’s what was meant when I introduced this notation in this post of mine:

@Magnus Anderson

Every whole decimal number can be represented in the following way:
(d_n \dotso d_3d_2d_1)

Here, (d_n) represents the leftmost digit whereas (d_1) represents the rightmost digit.

And this is by no means something new. If you go to Wikipedia, you can see a similar notation. The only difference is they use (a) instead of (d) (which I prefer because it’s short for “digit”) and they start with (0) instead of (1).

So the question is: what exactly is your problem?

But you didn’t. And worse, when you tried - you cast blame on the prior for being “wrong”. That is what they call “virtue signaling” - “Our new way is RIGHT and GOOD and any other way is WRONG! and EVIL!”

Stop ranting about politics and start explaining what’s wrong. Or just shut up.

[Magnus_Anderson]

@obsrvr524

But you didn’t. And worse, when you tried - you cast blame on the prior for being “wrong”. That is what they call “virtue signaling” - “Our new way is RIGHT and GOOD and any other way is WRONG! and EVIL!”

What “prior”? Who cares about what came before?

The Wikipedia approach is more elegant for very simple reason. The index of every digit is the same as the exponent associated with it. That’s simplicity. It’s easy to memorize.

In your approach, the index of every digit is different from the exponent associated with it. Just look at it:

(d_1d_2d_3 \dotso d_n = d_n \times 10^0 + d_{n-1} \times 10^1 + d_{n-2} \times 10^2 + \cdots + d_1 \times 10^{n-1})

The exponent associated with (d_1) is (n - 1). The index of the digit is (1) but the associated exponent is (n - 1). Two different numbers.

[Magnus_Anderson]

@Magnus Anderson

1. we have to say that, in our case, neither (\infty) nor (\dotso) indicate “any number greater than every integer”, but rather, that both indicate one and the same specific number greater than every integer

@obsrvr524

Your (1) doesn’t seem to be true.

Although the symbol (\infty) can possibly indicate ANY infinite “number” (higher cardinality), the “…” ONLY refers to the entire natural number set. I don’t think anyone every uses “…” to refer to a higher cardinality.

That was NOT a statement about how OTHER people define those symbols. That’s why I said “in our case”. That was merely a proposal on how to define those symbols in certain situations so as to increase the chances of other people correctly interpreting what we’re saying.

On the other hand, I do not agree that “…” means “repeat for a number of times equal to the number of naturals”. I would like to know where you got that from.

The “…” isn’t about how to construct the symbol or representation. It is about representing the value.

I think “…” means “repeat endlessly”. The part that you should repeat is the one that precedes the ellipsis. In the case of “999…”, it means “repeat 999 endlessly”. Exactly what is meant by “endlessly” is not clear. What’s clear is that the number of repetitions must be greater than every integer. Whether it refers to any such number or something more specific isn’t clear.

That we disagree on. Again apparently you are focused far too much on the positioning of the symbols (much like Certainly true) - ignoring the relevant meaning of those symbols.

Or it might be the case that you are not paying enough attention to the positioning of the symbols, failing to realize that the meaning of (999\dotso) is decided by its digits and how they are positioned.

Do you agree that if two decimal numbers don’t share the same exact digits, that they are not equal?
(That’s also why (1 \neq 0.\dot9).)

If so, it’s important to compare (999\dotso) to your number to see if they share the same exact digits.

Do they?

Notice that (999\dotso) has the most significant digit. Your number, on the other hand, which is (\sum_{i=0}^{i->\infty} 9\times10^i), does not. It has the least significant digit (evident in the fact that (i) starts at (0)) but it does not have the most significant digit (evident in the fact that (i) tends towards (\infty) without ever reaching it.) That’s precisely the opposite of (999\dotso) which has the most significant digit but no the least significant digit.

the “…” cannot reasonably be placed before the string of digits that define what is to be repeated

Why not? In the case of (\dotso999), it means “repeat the part that comes after the ellipsis, repeat it to the left and repeat it endlessly”.

[Great_Again]

@obsrvr524

@Great Again

@obsrvr524

I think he should have said - »earnestly seek and ye shall find«. Disingenuously evade the evidence and you merely lose sight of the Truth - but still have to face Reality".

Is this the edit you meant?

Yes.

@Great Again

I can agree with it, although there is a a little problem with finding.

Finding is similar to the epistemological subjectivity/objectivity problem, a duality. But nevertheless I agree to the said sentence “earnestly seek and ye shall find” and the following sentence.

What you said about truth and reality - “you can hide from one but never escape the other” is even more agreeable, regardless whether I would use the word “facts” instead of the word “reality” as well. Both refer to what we perceive, namely what has become, that is, what is completed through history. So, I would use the word “facts” as well as the word “reality”.

It is right what you said about truth and reality: “you can hide from one but never escape the other”. If you try to escape from truth, you have to face the reality (facts); if you try to escape from reality (facts), you have to face the truth. You are not as fast as the both are. Lies have short legs; unreality is like a dream in which movement is impossible.

Actually I meant the reality of the future consequences due to the reality of the past events. If I say, “facts”, I just throw in more opportunity of get into the semantics of “who’s facts”.

And even though the following seems off topic here, it is actually related (to why bother pursuing these words and what they mean).

When a society tries to govern by establishing an artificial truth narrative (such as real world Communism and those trying to govern the US right now), they create an artificially induced public bubble of belief and force the issue of having to stop all thinking. Reasoning tends to unveil the reality that it is all artificial - creating doubt in the veracity of the narrative.

I think establishing an artificial reality narrative for a population can be good or bad but is most certainly a very, very dangerous thing (as has been proven over thousands of years of attempts). It stems from the lust to advance by trying to be God - in place of God (in place of Reality). And they actually know that it is extremely dangerous - to everyone else. They fully know that they are going to murder billions of people - they simply don’t care. To them being a god over all life is just too blindingly important. And they know that you can’t catch them. So they proceed.

Those kind of people seem to think that they actually have no choice. They think that the only way to advance is through the means they currently see - their “path to godhood”. And the reason they think that is because of what we are discussing on this very thread -

Getting the real story straight before making presumptions about what is necessarily true - look CLOSER at the details of your own thoughts and language (which largely affects your thoughts). Seek out, not merely A path to victory, but the BEST path at all possible (assuming you are not in a rush).

The problem is that it is an ethical issue. And when people, high or low, simply don’t care - they don’t bother (relating to James’ PHT). I think that on high they are very willing to murder billions of people because they haven’t bothered to seek out a more ethical solution to advancing humanity toward a more harmonious existence on Earth (another seriously big issue with James - “Saint”).

I can see where he is coming from. I started by merely being curious - then impressed - then amazed - then awestruck - then just speechless (as you said - “impressed with him” was only in the beginning). I have been closely examining his proposals and what I call his “final solution” (his SAM Co-op) for the possibility of any tiny little misguiding hidden devil. I haven’t concluded that his final solution is pure and more importantly that it is the BEST solution to human harmony. I think that I have just recently realized who he was talking to all that time (a serious question I have held) - and it was not anything I was guessing. His solution seems to be a calculated far off destiny - and AFTER billions of people have been murdered - exterminated by what he referred to as the source of all of the trouble - the blind lust of the “Godwannabes”.

I have nothing better to do besides my wife, work, and wealth pursuits than to see if his unfortunately distant solution, possibly after the death of literally all human life, is really the BEST possible. And that requires a degree of examination for that elusive devil in the details that a “normal” person ( ) would never bother to find.

In short - details matter in thinking and trying to solve real world problems involving humanity - not merely maths or scientific war weaponry - “Get the words straight”

Well said.

[wtf1 (wtf)]

@Magnus Anderson

Though I do agree that definitions are necessary, I do not think that the existing definition of the word “size” only applies to finite sets. I think it applies to sets in general. That said, there’s no need to come up with a separate definition. We just have to deduce properly.

We’re mostly in agreement on what you said in this post.

@Magnus Anderson

An argument in favor of the idea that the notion of size is defined with respect to sets in general is the fact that infinite sets are defined by how many elements they have. We say that a set is infinite if its number of elements is greater than every integer. (To say that something is infinite is to specify its size. If you say that a set is infinite, you’re saying that the number of its elements is NOT an integer. The word “infinite” is not like words such as “red” that say absolutely nothing about the quantity of something.)

Well not that, but nevermind.

@Magnus Anderson

The set of naturals is greater than the set of even naturals for the very simple reason that it has all of the elements that the set of even naturals has plus some more.

I perfectly well agree if we focus on the proper subset relation. That’s one way of talking about a set’s size. Cardinality is another. Natural density (which I think I mentioned to you earlier) is yet another. Which is why “size” is ambiguous unless you carefully qualify it.

@Magnus Anderson

The fact there’s a bijective function such as (f(x) = 2x)

Hey man thanks, can you please explain that to my friend @obsrvr524? He’s a little confused on this point.

@Magnus Anderson

between them cannot change that fact. (At best, it can indicate a contradiction, that some sets are both equal and not equal in size.)

I perfectly well agree that by cardinality, the naturals and the evens are the same size; and by the proper subset relationship, the evens are smaller. That’s not a contradiction, it’s simply a failure to properly disambiguate between size(\text{proper subset}) and size(\text{cardinality}).

Note that the proper subset and cardinality do not play well together. The naturals have the same cardinality as the multiples of 4, via the bijection f(n) = 4n. But the multiples of 4 are a proper subset of the evens. So by proper subsets, mults of 4 < evens < naturals; but the mults of 4 and the naturals are in bijective correspondence. In general, cardinality tends to be the more useful measure of set size, which is why it’s so commonly used. But as long as you are careful to say that you are using the proper subset relation, I have no problem with that. But the thing is, you can’t get too far logically with that idea for the reason I mentioned, that it conflicts with cardinality.

@Magnus Anderson

What you can do is change the definition of the word to mean something else.

We just need to be precise about which meaning of size is being used at any given time.

The main point on which you and I disagree is that the notation (10^\infty) is not defined and has no meaning; whereas you are casually using it as if it did.

[Certainly_real]

@obsrvr524

I just have to hope that it isn’t intentional.

On my part, my intentions are clear. I am trying to provide/gain clarity on something that I clearly see as needing to be clarified.

What I see is a distinction without a difference.
But actually in A there are no fractions.
And in B there are only fractions (or actually decimals).

Are we in agreement that there is a clear difference between the following:

A) That which goes on forever 1, 2, 3, 4, 5…
B) The Set of ALL numbers

That which goes on forever is necessarily NOT the set of all numbers. You cannot count to infinity, thus you cannot exhaust the set of all numbers by counting forever 1, 2, 3… Do you see how B encompasses A but A does not encompass B?

B is what Infinity encompasses/contains/denotes. B contains A and IS more than A. Do you see why I made that distinction before? The following is CRUCIAL:

If I say the Set of ALL triangles, again, I am referring to Infinity in quantity. If I say the Set of ALL numbers/squares/fractions/words/humans/unicorns/ (any thing that is meaningful), I am referring to Infinity in quantity (the SET of any true ALL). Give it some thought. If I limit all to a finite context, then I am no longer truly referring to ALL. For example, if I say the set of all numbers or sticks in this box, then I am not referring to the set of ALL numbers or sticks. Just a limited set.

There’s a reason why Russell’s paradox and Cantor’s paradox are linked. ZFC solved nothing. It just ignored the problem.

[Certainly_real]

@Magnus Anderson

@obsrvr524

What word would you use to describe this kind of number - 0.999… ?

@Certainly real

Infinity

@wtf

Can you explain your reasoning for this? I think .999… = 1. But even if someone disbelieves that, they are going to say that .999… is just a smidge less than 1. People tend to think it’s “smaller than 1 by an infinitesimal amount.”

1 is finite so a quantity a tiny tiny bit less than 1 is still finite. I don’t follow why you think this is infinite.

If we take it that what Certainly real means by “infinity” is “a number greater than every integer” then (0.\dot9), being smaller than some integers, is not a number greater than every integer, and thus, not “infinity”. (Whether we take it that (0.\dot9) is equal to (1) or less than (1) has no impact on that and thus it’s irrelevant.)

But is it a finite number? I can’t tell because I don’t know what that means. What exactly is a finite number?

I think it’s a finite that continues to increase endlessly. It is not Infinity.

If you start counting natural numbers and go on forever, 1, 2, 3, 4… , you will not count to Infinity. Thus you will not exhaust/encompass ALL natural numbers.

If however you said x is the set of ALL natural numbers 1, 2, 3, 4… , you are then necessarily referring to Infinity. Only Infinity exhausts/encompasses ALL natural numbers.

Change ‘natural numbers’ to ‘even numbers’ and apply the above. You will get the same result.

If you say 1, 2, 3, 4… (meaning that the numbers go on forever), then your set is a finite set that continues to increase in size forever. It is not an infinite set.

If you say 0.999… (suggesting that the 9s go one forever), you are not highlighting an infinitesimal. You need an Infinity of 9s to highlight an Infinitesimal.

Confusion could stem from not distinguishing between an Infinity of 9s, and 9s that just go on forever. Again, an Infinity of 9s, encompasses/exhausts 9s that just go on forever. The latter is not Infinite. Thus, it does not denote Infinitesimal (because it does not amount to Infinity in any way). The former, encompasses the latter because it amounts to Infinity (an Infinity of 9s).

[wtf1 (wtf)]

@Certainly real

Are we in agreement that there is a clear difference between the following:

A) That which goes on forever 1, 2, 3, 4, 5…
B) The Set of ALL numbers

That which goes on forever is necessarily NOT the set of all numbers. You cannot count to infinity, thus you cannot exhaust the set of all numbers by counting forever 1, 2, 3… Do you see how B encompasses A but A does not encompass B?

Yes, this is a good insight.

The Peano axioms give you all the numbers 0, 1, 2, 3, 4, …

But they do not give you a set containing all of them. For that, you need the axiom of infinity in ZF, which says that there is a set that contains all the natural numbers. Technically it says that there’s a set that contains the empty set, and whenever it contains a set (n), it also contains the set (S(n) = n \cup {n}). Then by defining 0 as the empty set and S(n) as n+1, we see that we have a model of the Peano axioms in ZF; along with a set containing all the numbers created by the Peano axioms.

The axiom of infinity is stronger than the Peano axioms ¶, as you note. The reason mathematicians prefer to work in ZF rather than PA is that it’s difficult if not impossible to get a decent theory of the real numbers off the ground in PA. I’ve never seen it done. The axiom of infinity lets you use the set of natural numbers to build up the integers, rationals, and then the reals. And the reals, even though they’re purely a mathematical abstraction, are nonetheless important to physicists, who study the real world. So there is something important, even if we can’t put into words, in our conception of the set of natural numbers.

But then again from a philosophical standpoint, I don’t see why the axiom of infinity is problematic. I know there are giraffes at the Bronx zoo, and I can conceive of the entire collection of them, even though I haven’t been to the Bronx zoo in years and don’t remember seeing the giraffes last time I was there. Likewise having conceived of the sequence 0, 1, 2, 3, 4, …, my mind has no trouble conceiving of the collection that contains all of them.

[Certainly_real]

@wtf

@Certainly real

Are we in agreement that there is a clear difference between the following:

A) That which goes on forever 1, 2, 3, 4, 5…
B) The Set of ALL numbers

That which goes on forever is necessarily NOT the set of all numbers. You cannot count to infinity, thus you cannot exhaust the set of all numbers by counting forever 1, 2, 3… Do you see how B encompasses A but A does not encompass B?

Yes, this is a good insight.

The Peano axioms give you all the numbers 0, 1, 2, 3, 4, …

But they do not give you a set containing all of them. For that, you need the axiom of infinity in ZF, which says that there is a set that contains all the natural numbers. Technically it says that there’s a set that contains the empty set, and whenever it contains a set (n), it also contains the set (S(n) = n \cup {n}). Then by defining 0 as the empty set and S(n) as n+1, we see that we have a model of the Peano axioms in ZF; along with a set containing all the numbers created by the Peano axioms.

The axiom of infinity is stronger than the Peano axioms ¶, as you note. The reason mathematicians prefer to work in ZF rather than PA is that it’s difficult if not impossible to get a decent theory of the real numbers off the ground in PA. I’ve never seen it done. The axiom of infinity lets you use the set of natural numbers to build up the integers, rationals, and then the reals. And the reals, even though they’re purely a mathematical abstraction, are nonetheless important to physicists, who study the real world. So there is something important, even if we can’t put into words, in our conception of the set of natural numbers.

But then again from a philosophical standpoint, I don’t see why the axiom of infinity is problematic. I know there are giraffes at the Bronx zoo, and I can conceive of the entire collection of them, even though I haven’t been to the Bronx zoo in years and don’t remember seeing the giraffes last time I was there. Likewise having conceived of the sequence 0, 1, 2, 3, 4, …, my mind has no trouble conceiving of the collection that contains all of them.

Thank you. I didn’t study maths. I studied philosophy. I tried to make sense of the above and the link, but I struggled (except for the part about it guaranteeing at least one infinite set, which I think is a good move). There is one clear problem I have with ZF. The following is from Wikipedia:

Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets)

To me, that clearly says ZF set theory is contradictory.

I am familiar with Frege’s definition of an empty set from philosophy. The set of all necessarily meaningless things like round squares or married bachelors. The set is empty because it contains nothing meaningful. Or perhaps you can say it contains ALL contradictory semantics. In trying to understand the above, I didn’t get how an empty set works in maths. I would have guessed it would contain something like the square root of the cardinal 2. No two identical cardinalities can be multiplied together to result in the cardinal 2. So I think the square root of the cardinal 2 is absurd. So it should be in the empty set. But I’m definitely projecting philosophy here onto a maths issue.

[wtf1 (wtf)]

@Certainly real

Thank you. I didn’t study maths. I studied philosophy. I tried to make sense of the above and the link, but I struggled (except for the part about it guaranteeing at least one infinite set). I am familiar with Frege’s definition of an empty set from philosophy. The set of all necessarily meaningless things like round squares or married bachelors. The set is empty because it contains nothing meaningful. Or perhaps you can say it contains ALL contradictory semantics. In trying to understand the above, I didn’t get how an empty set works in maths. I would have guessed it would contain something like the square root of the cardinal 2. No two identical cardinalities can be multiplied together to result in the cardinal 2. So I think the square root of the cardinal 2 is absurd. So it should be in the empty set. But I’m definitely projecting philosophy here onto a maths issue.

You have it right. The empty set is the set that doesn’t contain anything. But if you like, you can (and I often do) think of the empty set as the set containing all the purple flying elephants in my left pocket. That’s the same as your idea.

But the details aren’t important. What’s important is this:

The Peano axioms give us each of 0, 1, 2, 3, 4, 5, …

ZF including the axiom of infinity gives us the set {0, 1, 2, 3, 4, …} that contains all of the numbers given by the Peano axioms.

So you are correct that ZF is stronger than PA. PA merely gives us a bunch of groceries. ZF gives us a grocery bag to put them in. And the empty set is just an empty grocery bag!

@Certainly real

Specifically, Zermelo–Fraenkel set theory does not allow for the existence of a universal set (a set containing all sets)

No, it’s just that if we assume the axioms of ZF, then the assumption that there is a universal set leads to a contradiction. Since we prefer to have a set theory free of contradictions, we reject the universal set. I suspect you know this, it’s Russell’s paradox, which is a straightforward argument. And you know that once Frege saw the paradox, he recognized its correctness. I assume that should carry some weight with you.

If you’d like an intuitive argument, it’s that the word “set” is much more restrictive that we think. A set is much more than just a collection or assemblage or plurality. A set is a collection or assemblage or plurality that satisfies the axioms of set theory. And those axioms put constraints on what a set can be.

@Certainly real

To me, that clearly says ZF set theory is contradictory.

ZF plus “there exists a universal set” is contradictory. So we throw out the universal set; and we are left with ZF, which as far as we know is not contradictory.

[obsrvr524]

@Certainly real

@obsrvr524

I just have to hope that it isn’t intentional.

On my part, my intentions are clear. I am trying to provide/gain clarity on something that I clearly see as needing to be clarified.

I think I can acknowledge that. But similar to Mr Anderson, I think you have map vs terrain issues.

@Certainly real

Are we in agreement that there is a clear difference between the following:

A) That which goes on forever 1, 2, 3, 4, 5…
B) The Set of ALL numbers

That which goes on forever is necessarily NOT the set of all numbers. You cannot count to infinity, thus you cannot exhaust the set of all numbers by counting forever 1, 2, 3… Do you see how B encompasses A but A does not encompass B?

No - not really.

Again, “that” which goes on forever - is a “that” - a single thing. There is no counting involved except in the category description of what “that” is to represent. “That” in itself has no counting involved.

NONE of what we have been discussing has anything to do with actual counting or the ability to get to an end. In every case represented we have been talking about “what you would get IF all was counted” - the final sum - the one quantity or size of the total.

The descriptive representation is NOT the value itself - only an effort to explain what single final total is being presented. And that can be called a “set” or a “sum” or “ALL” or even “999…” or “0,1,2,3,…”. They are ALL presenting the idea of a single thing - one - no counting - no need for the ability to get to an end. The “end” (when there is one) is already there. When there is no end, it is called an “open” or “unbound” set - but still a set none the less.

Infinity is a different issue - the first successor location that is just past the end that isn’t there.

And you can have a universal set. You just can’t have a set of all sets (a Dedekind set).

[Magnus_Anderson]

@wtf

We’re mostly in agreement on what you said in this post.

I perfectly well agree if we focus on the proper subset relation. That’s one way of talking about a set’s size. Cardinality is another. Natural density (which I think I mentioned to you earlier) is yet another. Which is why “size” is ambiguous unless you carefully qualify it.

I perfectly well agree that by cardinality, the naturals and the evens are the same size; and by the proper subset relationship, the evens are smaller. That’s not a contradiction, it’s simply a failure to properly disambiguate between size(\text{proper subset}) and size(\text{cardinality}).

Note that the proper subset and cardinality do not play well together. The naturals have the same cardinality as the multiples of 4, via the bijection f(n) = 4n. But the multiples of 4 are a proper subset of the evens. So by proper subsets, mults of 4 < evens < naturals; but the mults of 4 and the naturals are in bijective correspondence. In general, cardinality tends to be the more useful measure of set size, which is why it’s so commonly used. But as long as you are careful to say that you are using the proper subset relation, I have no problem with that. But the thing is, you can’t get too far logically with that idea for the reason I mentioned, that it conflicts with cardinality.

We just need to be precise about which meaning of size is being used at any given time.

I see. So your position is that there are two different notions of size that we must be careful not to conflate. According to one, we say that any two sets are equal if and only if there exists a bijective function between them. With that definition in mind, the set of naturals and the set of evens are of the same size because there exists a function such as (f(x) = 2x) that maps the set of naturals to the set of evens. I have no problem with that.

Where we might disagree is the following:

1. I am not sure that other people (mathematicians, in particular) see it this way (at this point in time, I have no argument to present in defense of this belief; I will need time to collect my thoughts and come up with one)

2. I believe the other notion of size, the one that I am working with, precedes the notion of size qua cardinality (it’s not something that I invented nor something that was invented after the notion of cardinality)

3. I believe that ambiguity is created by the usage of the word “size” to mean something other than what it normally means

4. Though I do agree that cardinality is useful, I do not think that the other notion of size is of little to no use

@wtf

The main point on which you and I disagree is that the notation (10^{\infty}) is not defined and has no meaning; whereas you are casually using it as if it did.

I don’t know why you think that (10^{\infty}) is not defined.

Can you help me with that?

(10^{\infty}) is another way of expressing (10 \times 10 \times 10 \times \cdots).

And (\infty) means “a number greater than every integer”. In other words, it’s a symbol that can be used to represent any number greater than every integer.

Thus, (10^{\infty}) means “(10) raised to the power of a number greater than every integer”.

It’s akin to (10^x) where (x) is defined to mean “a number greater than (3)”. In such a case, (10^x) would mean “(10) raised to the power of a number greater than (3)”.

[Magnus_Anderson]

en.wikipedia.org/wiki/Cardinality

@Wikipedia

Two sets A and B have the same cardinality if there exists a bijection (a.k.a., one-to-one correspondence) from A to B,[5] that is, a function from A to B that is both injective and surjective.

If what they mean by one-to-one correspondence is that the elements of (A) and (B) can be paired in such a way that every element in (A) is paired with exactly one element in (B) and every element in (B) is paired with exactly one element in (A), then there is no one-to-one correspondence between the set of naturals and the set of evens.

The reasoning is super simple and I am sure that you, wtf, are aware it.

The set of naturals contains every element of the set of evens plus some more. Thus, once you pair every even number with a unique natural, you are left with a lot of unpaired naturals.

On the other hand, I do agree that there is a bijective function between the two sets.

Wikipedians seem to think that bijective function and one-to-one correspondence are one and the same thing:

en.wikipedia.org/wiki/Bijection

@Bijection

In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements.

So one of the following must be the case:

1. they do not use the term “one-to-one correspondence” the same way as I do above

2. the term “bijective function” does not mean what they think it means

3. there is a contradiction

Which one is the case? Or maybe I left out some other possibility?

[obsrvr524]

Again -

@obsrvr524

Let’s say that N = the natural number set = {0,1,2,3,4,5…}
And that B is N reordered to {0,1,3,4,5…,2}.

It is easy to see that N bijects with B because both sets are equality countable = equinumerous.

But what about set C = {N, a, b, c}

Do you believe N bijects with C?
Is C even countable?