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.
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.
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.
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).
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).
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.
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.
Two issues -
Why do you keep saying “non-zero digits” when ALL of the digits are clearly non-zero - “9”
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.
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.
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?”
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).
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.
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).
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.
You are supposed to explain what I am doing wrong instead of ranting about politics.
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:
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?
Stop ranting about politics and start explaining what’s wrong. Or just shut up.
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:
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.
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.
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.
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”.
We’re mostly in agreement on what you said in this post.
Well not that, but nevermind.
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.
Hey man thanks, can you please explain that to my friend @obsrvr524? He’s a little confused on this point.
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.
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.
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.
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 Infinityin 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.
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).
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.
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!
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.
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.
I think I can acknowledge that. But similar to Mr Anderson, I think you have map vs terrain issues.
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).
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:
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)
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)
I believe that ambiguity is created by the usage of the word “size” to mean something other than what it normally means
Though I do agree that cardinality is useful, I do not think that the other notion of size is of little to no use
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)”.
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: