Is 1 = 0.999... ? Really?

Philosophy Forums Science, Technology, and Math
781

No worries. I lost my mind years ago. And just ask Saint about my credibility.

You kind of lost me here. Any decimal expression is a function from the naturals to the digits. Although it’s more convenient to just consider numbers between 0 and 1 so you don’t have to work out the encoding of the digits to the left of the decimal point. I’m not sure what kind of function you’re trying to work out here. There’s only one function from the naturals to the number 1, namely the identity function that maps everything to 1. I’m kind of lost here.

I’m lost totally. If you are saying that 1 isn’t 1.000… I’m really confused.

However it is the case that in set theory the natural number 1 is not the same set as the real number 1. This leads to discussions of the philosophy of structuralism in mathematics. Is 1 a particular set? Clearly not, since many different sets can represent it. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics)

I’ve never heard of it. It’s not an axiom of set theory. What is it?

Yes of course. ZFC is the standard set theory that rules all of contemporary math except for the areas that have been taken over by category theory (algebraic geometry for example, including Wiles’s proof of FLT) and advanced set theory in which all kinds of weird alternative models are considered.

However: I know of no set theory in which .999… isn’t 1! That’s sort of the point. it’s very hard to come up with a logically consistent theory of math in which .999… is anything other than 1. Even in the hyperreals, where there are lots and lots of infinitesimals, .999… = 1.

That is certainly a valid philosophical point. ZFC won the 20th century but of course it’s nothing more than a historically contingent idea. Archimedes, Newton, Gauss, Euler, and a lot of other people did brilliant math without ever having heard about set theory. Set theory has lots of flaws. One of them being that the natural number 1, the integer 1, the rational 1, the real 1, and the complex 1 are all different sets but the same number.

These days people are into category theory, intuitionist type theory, homotopy type theory, alternative set theories, all kinds of other stuff.

What do you mean by homomorphism? The naturals aren’t a group. Do you mean bijection? The evens and the odds are in bijective correspondence but they’re not equal as sets. Can you clarify the question?

782

That is not the definition of “infinite”. That is an inference using flawed supposition.

That is about as ridiculous as one could get and completely false.

I don’t know who you are reading, but it obviously isn’t me.

I saw it being used recently and have seen it often in the past.

I have ask for a proof that bijection works for infinite sets. I know that wft can’t find or post proofs. I was hoping that Careas might come up with one. Maybe Ed3?

783

I’ve asked you to clarify what you’re looking for in such a proof. A set having exactly one element for every element in another set (again, “exactly one”) seems to be the equivalent of saying that the two sets have the same number of elements, i.e. the same cardinality.

784

Interesting topic. Definition of infinity. The modern definition, that a set is infinite if it is in bijection with one of its proper subsets, is due to Richard Dedekind.

Wikipedia says:

Now you do have a valid point that there are a number of other definitions of infinite sets; and there are some subtle logical differences among them.

If you would like to propose an alternate definition of an infinite set, I’m open to hearing about it.

Don’t hold back. Tell me what you REALLY think!!

Someone using your handle (do you have a cat?) wrote

If you can clarify exactly what you are asking, it will be helpful.

You could never have such a thing. It’s like asking for a proof that a triangle is a three-sided polygon. You can’t prove that because triangle is the DEFINITION of a three-sided polygon.

Please explain what you mean by “bijection works for infinite sets.” Given two infinite sets there either is or isn’t a bijection between them. If there is, we say they have the same cardinality. It’s no different than saying that if a polygon has three sides we call it a triangle.

I wonder what you mean by that, since I’ve posted or linked many proofs in this thread. I will grant you that what a classical logician regards as a proof is not the same thing as what a mathematician regards as a proof. That’s what I think you mean.

In fact if a proof is what it’s defined to be in mathematical logic; then no working mathematician has ever seen a proof! You are absolutely right about that. We can talk about this sometime if you’re interested.

Nobody will ever be able to prove a definition. You can’t prove the wet stuff coming from the sky is rain. The wet stuff coming from the sky is CALLED rain. That’s its name. When two sets have a bijection between them we SAY they have the same cardinality. Cardinality is the name we give to that property. Unlike with the finite numbers, where we can use our fingers to define 5, cardinality for infinite sets can ONLY be defined in terms of bijections. That’s the thing we found that works!

I wish you’d explain your point better. I know you’re trying to express something but I’m not understanding it. When you ask for a proof that "bijection proves equality of cardinality for infinite sets " – a direct quote of your words – I don’t know what you mean. Bijection is the DEFINITION of equality of cardinality for infinite sets.

785

And I have answered your questions every time.

That is what has been intuited. But as you know, intuition is often found to be wrong. And this happens to be one of those cases. I can prove why it isn’t true for infinite sets. But it would be better if you provided what you believe to be a reasonable proof first so that it will be more clear as to why your intuition fails you.

786

I’d like to see such a proof. For anyone’s definition of proof.

787

So you found one typical philosophically naive mathematician making claims and redefining another word for his convenience. He cannot redefine a word that is so very common and already defined without creating ambiguity. Mathematicians and physicists don’t understand that, but its quite true (Hell, ask any English teacher).

Simple;
Infinite == endless, unbound, not closed (Duhhhh-uh).

I have seen your attempts at proofs. The word “clueless” comes to mind. Go back to sleep.

See. There you go. Let the more philosophically enlightened handle this.

788

I think it’s about time to wrap this one up: Does or does not 1 = 0.999?

789

Definitionally … not.

790

In what particular context?

791

Well, this time I am willing to admit that dasein, conflicting goods and political economy have nothing to do with whether 1 = 0.999 or not.

They don’t, right?

Mathematics would seem to be above all of that godawful “is/ought” stuff.

Either it is or it is not. And the fact that these complex exchanges here seem to revolve around the question as a philosopher sees it and as a mathematician sees is, rather confound folks like me who do not have a background in mathematics.

We’re thinking: Can this be resolved or not?

All the while thinking that with mathematics this sort of thing would seem to be considerably less likely to be left unresolved.

792

I have not seen a valid philosophical objection.

What I’ve seen is that whenever I accept JSS’s premises, follow them to their logical conclusion and frame a question, JSS hurls some insults and changes the subject.

He keeps talking about logic. So I pointed out that the logic of modern math is first-order predicate logic. I asked him if that might be a source of our disagreements. He ignored the question and hurled some insults.

I’ve asked him half a dozen times why he even cares about the hyperreals when .999… = 1 is a theorem of the hyperreals anyway. He’s never engaged on that point.

He’s said that he disagrees with me on what a proof is, so I pointed out that there is a difference between what a logician and a philosopher regard as a proof. I specifically offered to discuss the subject. I’m prepared to discuss the history of the nature of mathematical proof from antiquity to the present. He ignored the point and hurled more insults.

Is that what you mean by a difference of opinion between a philosopher and a mathematician? Isn’t JSS basically an ignorant troll?

793

Okay, from your point of view, definitionally, 1 does not equal 0.999. How then would this be the same or different from arguing that President Trump is guilty or innocent of obstruction of justice in fact or guilty or innocent of obstruction of justice “definitionally”?

When would the two be exactly the same?

We can take this to another thread if you’d like.

794

Continue and we are going to discover quite the opposite.

So you say that all of your dasein and is/ought concerns have nothing to do with math (as they don’t with RM:AO either), yet here you are injecting the guilt or innocence of a politician as something relevant?

795

…

796

I have said several times that your categories are not as neatly separated as you think. This is one example.

It seems that all rational men and women are NOT obligated to think one way about 1=0.999…

It seems that it can’t be demonstrated adequately one way or the other.

Those tiny numbers, infinitesimals, also show how “might makes” right worked in mathematics. Infinitesimals were critical to the discovery of calculus but they were not rigorously defined and the faction which supported “limits” gained control in academia - infinitesimals because a fringe idea. Yet society could have gone another way and adopted a number system which contained infinitesimals. It’s even possible for a society have advanced mathematics and not to use the decimal system at all - the 1=0.999… would not arise at all.

Mathematicians argue about this as well.

Mathematics is not entirely ossified. There are advancements, and there are changes in the way we use math and why we use it.

The “non-standard analysis” of today may become the “standard analysis” of tomorrow.

797

If that is a theorem, nobody is stopping you from posting it or linking to it.

798

What he has been referring to is the theorem that what is true for the reals is also true for the hyperreals and [since 1 = 0.999… in the reals] 1 = 0.999… in the hypereals, therefore 1 = 0.999… in the reals - presumptuous circular reasoning.

799

This is bullshit, James. You’re discussing in bad faith. “I could offer argument or evidence in support of my position, but I won’t because I’d rather see you try to defend your and then ridicule your attempts” is a fucking bullshit way to engage someone else in a conversation.

800

I have given the proofs several times already in this thread. But it’s far more sophisticated than the countability of the rationals, and you’re not willing to work through that. You said so yourself.