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This question is somehow of a follow up to to this other one, and it's something that has bugged me for a while.

I understand the notion that there's no "absolute truth" in math, in the sense that every theorem follows from an assumed set of axioms. The typical example is euclidean vs non-euclidean geometries.

The question I linked has an comment asking for clarification on the notion of an absolute truth. The way I understand the distinction between an non-absolute (claim of) truth and an absolute one is this: A non-absolute claim C is really just a short way of claiming: ZFC (or some other set of axioms we all agree upon) implies C. And an absolute claim C just claims C and nothing else.

So here is the thing that bugs me, if the claim 1+1=2 is just a short way of claiming "ZFC implies that 1+1=2" then is this claim itself absolute, or is it really claiming something like "XYZ implies (ZFC implies 1+1=2)"? Or, to put it another way, if you prove to me that 1+1=2 using ZFC axioms, and then I ask you to prove that your proof is correct, which axioms would you use?

I guess one option would be for XYZ to be just ZFC itself, so that "ZFC implies (ZFC implies 1+1=2)", but that's just equivalent to "ZFC implies 1+1=2", which would make the claim an absolute one, given the distinction I made above. The alternative is to have XYZ refer to some other set of axioms, but then either this claim would be absolute, or we'd need yet another set of axioms, ad infinitum.

Is there a simple answer to this, or is this a topic I can dig some more about? I honestly don't even know how to google this specific question (as opposed to the one I linked). Or am I just missing something obvious and the question makes no sense?

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  • Saharon Shelah has spoken of "logical dreams" that would, I think, allow us to infer different arithmetic via forcing (I think), if the dreams could be realized somehow. Others think arithmetic represents an intuitive kernel of mathematical knowledge, and that we are more confident in intuitive arithmetic than any ZFC-like theory that seeks to obviate the appeal to intuition. Kant infamously thought that arithmetic was not logically necessary, hence not logically absolute. Jul 19, 2022 at 1:00
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    Given Von Neumann universe cumulative-hierarchy interpretation as a class model of ZFC we immediately have V0=∅, V(n+1)=P(Vn) ⊢ V1={∅} ∧ V2=P(V1)={∅,{∅}}, then we interpret V1 as 1 and V2 as 2 thus 2=1+1 is well-defined in ZFC. Of course the same is commonly defined in PA using successor function S. So 2=1+1 is actually not a theorem of ZFC (or PA) but just a true analytic a priori proposition per logical positivists (or 2 is just a term well-defined in V). But due to vacuous truth your conditional could be absolutely true within FOL... Jul 19, 2022 at 2:53
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    Gödel developed a technical notion of informal "absolute provability", as did Post, Tarski, Cohen, etc., see Leach-Krouse's thesis. But if you mean the philosophical sense of "absolute" then no, ZFC ⊢ 1+1=2 depends on human-all-too-human concepts, conventions, proof standards and practices, etc. All very non-absolute.
    – Conifold
    Jul 19, 2022 at 4:01
  • See also Leitgeb, On Formal and Informal Provability on Gödel's conception.
    – Conifold
    Jul 19, 2022 at 4:08
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    Not exactly the same, but have you ever heard the parable of Achilles and the tortoise? Jul 19, 2022 at 21:05

11 Answers 11

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Is the fact that ZFC implies that 1+1=2 an absolute truth ?

No. it is possible to create or imagine systems of logic in which the axioms of ZFC are true but 1+1=2 is false (or, at least, not provable).

A result (such as "1+1=2") only follows from a set of axioms (such as ZFC) if a set of rules of inference is given (such as "if A and (if A then B) then B"). But then you need a set of meta-rules that tell you how to interpret and apply the rule of inference. And then you need meta-meta-rules ... and you have an infinite regress.

This was famously pointed out by Charles Dodgson (writing as Lewis Carroll) in the article What the Tortoise Said to Achilles.

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    Not exactly. There's a proof that for some symbols x and y, there's an operation + that invariably results in x + x = y; and it turns out that x will always act as 1 and y as 2. But this proof doesn't actually operate on numbers but rather on some lower form. (The proof is over ZF; no C required. If you have a simpler set theory the proof can disappear. It's a set of axioms you don't have to take if you don't have ZF.)
    – Joshua
    Jul 19, 2022 at 17:59
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    OK I'm not sure if I agree that "no" is the actual answer. Given all the responses here it seems to be a bit more complicated. But the link you've provided is perfect, and I went from not even knowing how to google this to having this bunch of discussion and even a reply by Russell. So I'm marking this as the answer.
    – Juan
    Jul 19, 2022 at 23:04
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    To echo @Joshua, not so fast and not exactly. 1st, what Lewis Carroll has to say, as interesting as it is, has no bearing on what ZFC has to say about "1+1=2". You make a pretty bold claim that I'm pretty sure is not true, "it is possible to create or imagine systems of logic in which ZFC is true but 1+1=2 is false". Show us that system. Imagination plays no role in this. Without showing us (or linking to such a model someone else has established), your words are empty and meaningless in regards to what the OP is asking.
    – A.Ellett
    Jul 21, 2022 at 19:50
  • @A.Ellett I have edited my answer to be more precise. It is possible to create systems of logic in which the axioms of ZFC are true but 1+1=2 is false (or, at least, not provable). You simply change or omit some of the standard rules of inference. Now you could get round this by adding the standard rules of inference to the axioms of ZFC - but then you also have to add meta-rules, meta-meta-rules etc. It is this infinite regress that Lewis Carroll was pointing out.
    – gandalf61
    Jul 22, 2022 at 7:27
  • @gandalf61 I'd like to see a model of ZFC where 1+1=2 is not provable. If you "simply change or omit" rules of inference, then you're not really working with ZFC. But there is not infinite regress of meta-rules. The beauty of the mathematics developed around ZFC is that it is self contained. It can be completely see as rules for pushing symbols around and nothing more.
    – A.Ellett
    Jul 23, 2022 at 13:20
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Someone asked:

Isn't: 1 < 2, true everywhere?

My answer was thus:

In order to assign truth value to the proposition, one has to assign meaning to the symbols of the proposition. Else it is simply a string of symbols having no truth value in itself, except that it is true that it is a string of symbols.

So what does it mean "1", "2" and "<"? How does one assign meaning, that can apply to this world?

One approach would be to define "1","2", etc as the class of things that count to 1 or 2 and so on. Eg two hands, two feet, one head and so on.

So, all those things that have the same count act as instances of the class "2", representing 2.

What does it mean that "1 < 2" or that "1+1=2". We assign meaning to symbols "<". For this symbol, it means we assign an order, between numbers (and remember numbers are instances of objects in this world). We mean the number 1 comes before number 2 and so on, according to some well defined sense.

But, wait a minute.

There are sets of numbers that do not satisfy the total ordering principle

So how do we really know the set of numbers representing this world are of the one kind of set and nor the other?

Or even if they do, how do we know they continue to do so the next instant?

We don't! And this is where absolute mathematical truth ends.

So even though ZFC can imply 1+1=2, it can be true, false, or meaningless. Nothing absolute here.

As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.

Albert Einstein

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    The fact that some number sets cannot be ordered (like the complex numbers) has no bearing on whether the two natural numbers 1 and 2 can be compared. In fact, defining the ordering operator < is a vital step in defining the natural numbers (because you need to ensure that the +1 operator does not produce a circle). As such, as long as we are talking about the natural numbers 1 and 2, they are always ordered, even though they are also complex numbers which cannot be ordered in general. Because in this special case, they can. Jul 21, 2022 at 8:10
  • Sure, but one can conceive of natural numbers (eg modulo m) where ordering is not respected. The point is that we don't know what natural numbers are in the first place.
    – Nikos M.
    Jul 21, 2022 at 9:03
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    Not exactly. Once you turn to calculation mod m, you are not working with natural numbers anymore. You are working with a finite set of numbers, and thus need to redefine addition within that finite number set. This redefinition produces 1+1=0 in the case of mod 2, i.e. within the field of two elements. But since you didn't say that you do not use the natural numbers, the implicit assumption is that 1 and 2 are natural numbers, and thus that the statement must be true. Jul 21, 2022 at 9:15
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    Which world are they representing? Numbers are but an abstract construct. And there is a convention among mathematicians: Written numbers are natural, integer, rational, real and complex unless stated otherwise or not representable (i.e. 1 is a natural number, -1 is only integer, 1/3 is only rational, sqrt(2) is only real, and 1+i is only complex). You didn't state otherwise, so your 1 and 2 must be assumed to be natural numbers. And for natural numbers, the statement 1+1=2 holds. Jul 21, 2022 at 9:50
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    Math is independent of our world. And when you define something to be infinite, then it is infinite. Simple as that. The physical world has no say in this. Jul 21, 2022 at 15:12
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Yes, it is an absolute truth.

You are asking: "Is the fact that ZFC implies that 1+1=2 an absolute truth?"

Or equivalently: "Is the proposition ZFC implies that 1+1=2 an absolute truth?"

By definition (or conveniently, by Wikipedia), Absolute Truth (or "Universality") means:

In logic, [...], a proposition is said to have universality if it can be conceived as being true in all possible contexts without creating a contradiction.

So, the proposition "ZFC implies that 1+1=2" limits what contexts are possible, i.e. with regards to universality we only need to consider contexts where ZFC itself can be expressed at all.

The proposition then says that if ZF(C) is true in this context - in other words, if the ZF(C) axioms themselves are part of the axioms of our context, or if the context has axioms from which the ZF(C) axioms as we usually know them can be logically deduced, and if we assume the usual ZF mapping between the symbols "1", "2", "+" and the set-based ZF natural number creation mechanism - then a proof can be found that 1+1=2. (Not important, but the "proof" is of course very simple and basically just a direct application of the ZF set-theoretic definition of natural numbers.)

There may of course be other contradictions in the system, caused by the existence of more axioms which may be contradictory to ZFC, say. But those contradictions are there no matter what, and not created by our proposition.

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The word "absolute" itself is not absolute. "Absolute" is in opposition to "relative". It's generally easier to explain what "relative" means and then say that "absolute" means "not relative". A proposition P is relative if

  1. there is some set of possible contexts within which you evaluate propositions, and
  2. P is only true within a subset of those contexts.

An absolute proposition, Q, then is one such that

  1. Q is not evaluated with respect to such a set of contexts or
  2. Q is true (or false) in all such contexts.

The condition, "Q is not evaluated with respect to such a set of contexts" is needed because there is always some set of contexts you can insert after the fact to turn any proposition relative. This is true because no expression of any proposition contains all relevant details. There are always assumptions about what alternatives are being rules out by a proposition, and those alternatives can always be made explicit.

Mathematics is a good example of this. For the first few thousand years of mathematics, axioms (or assumptions, before Euclid) were considered to have no alternatives, so theorems were absolute. There were no possible alternative contexts. Then mathematicians discovered axioms systems as an abstraction. This meant that different theorems would be true in different axioms systems, so this meant that theorems are relative to the axiom system.

On the other hand, regular mathematics continues to proceed as if there were only one axioms system, and in these areas of mathematics, theorems are still absolute.

So, the answer to your question is: it depends.

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No proof in any formal system is a statement of absolute truth. It's a statement that given some axioms and some rules that we agree on, we can derive the validity of some proposition within the formal system. 1+1=2 is a formal statement, and it's validity relies on agreement about what '1', '+', '=' and '2' mean within a formal system. If we have such ageement, you cannot dispute the statement (provided it is valid within the rules) and if you don't have agreement, the statement is meaningless.

In other words, in order to be convinced that a statement in a formal system is correct, one must (for the sake of the exercise) agree to the axioms, rules and symbol definitions of the system.

That is a meta statement, not about a particular system, but about formal systems in general. So the next question would be "is this formal system well-formed"? There are also rules and definitions about that, and if those are agreed to, one should be able to demonstrate that a given formal system is well formed, and that therefore any valid proof in the system is well formed.

That is also not absolute. It depends on agreement about what makes a formal system well formed.

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    "agree to the axioms and rules of the system" plus the meaning of the symbols
    – Nikos M.
    Jul 19, 2022 at 5:38
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    OK, but this is a (better) restatement of what I said in my 2nd paragraph. My question would be what comes next (trying to use your words): Is the statement "if we agree on the axioms we cannot dispute 1+1=2" itself absolute, or does it rely on some other set of axioms and rules?
    – Juan
    Jul 19, 2022 at 9:24
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Mathematical axioms are not statements about the real world and therefore I think are better not thought of as having truth value in the sense you are saying. Let's look at 1+1=2 using the Peano axioms of arithmetic which will also hold in ZFC.

Define a successor operation S(x) and a number 0

Let 1 = S(0) and 2 = S(1)

Define an addition operation such that

  1. x + 0 = x and
  2. a + S(b) = S(a+b)

Now let's evaluate 1+1

By the definition of 1 this is equivalent to S(0) + S(0)

By the 2nd rule of addition above S(0) + S(0) = S(S(0)+0)

And by the 1st rule S(0) + 0 = S(0) so S(S(0)+0) = S(S(0))

And S(S(0)) = 2 by our definition of 2

So 1+1=2

Now there are no such objects as 0,1 or 2 in the observable world. The question is only whether or not in a given context there are objects or properties of objects that conform to the axioms that we have laid out (even the I have only laid out a subset of them for the purposes of the example). To the extent that there are objects or properties of objects in the real world which conform (more or less) to the axioms of arithmetic then we can use arithmetic to model them.

However, when we say that 1+1=2 I think we have good intuitions about the mathematical context we are dealing with and therefore I think it is fair to assert that this statement is in some sense absolutely true. While it is certainly true that this statement will not be true under a different set of definitions and axioms, that can also be said about any statement in any language. When we ask whether or not 1+1=2 we are not asking about the truth value of the syntax itself which of course has no inherent truth value, we are asking about the truth or falsehood of the proposition implied by that statement which includes all of the necessary context.

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    "Mathematical axioms are not statements about the real world [...]" might be worth adding "in my opinion", as there is a lot of philosophy against it.
    – user14511
    Jul 20, 2022 at 7:30
  • There's a lot of philosophy for it as well although I suppose that's a fair point that it's worth flagging the controversy. Still it seems that it must be true that not ALL mathematical axioms are statements about the real world since any mathematician could make up any structures and sets of axioms they want simply because they find them interesting. Jul 21, 2022 at 11:05
  • In other words given that whether or not your choice of axioms are or are not to be taken as statements about the real world is irrelevant from the perspective of doing mathematics. Therefore it's quite possible to construct a set of axioms which aren't motivated by the real world in any way. Therefore in the general sense mathematical axioms are not statements about the real world although that doesnt necessarily help answer the original question. Jul 21, 2022 at 11:10
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Note: I am interpreting your question in the precise sense of, "Is the conditional 'ZFC → 1 + 1 =2' absolutely true?" If this is a misinterpretation, nothing I am about to say applies all that much.

So now, though technically we can ask about conditionals being absolutely true, as far as the pragmatic reason for describing things as absolute goes, we would not tend to ask said questions. This is why Kant contrasted categorical with hypothetical imperatives, after all (even though in an extended sense, he framed, "I ought..." as an, "I would... if reason alone determined my will"). (So note that the Cantorian phrase absolute infinity might be reworded, modulo Kant, as categorical infinity, such that the alephs and the omegas are hypothetically or conditionally infinite.) (C.f. talk of "absolute in a domain" in a typical set-theoretic context.)

That being said, let us go to your algebra of theories XYZ (a variable over all possible such theories, let us suppose) and ask, "In XYZ [AKA any possible theory], is the conditional 'ZFC → 1 + 1 = 2' true?"

Perhaps not. Unless we hold that the system or method of logic "canonical" for ZFC application is also in play, perhaps we could use ZFC's axioms and derive something else, here. So the question might be refined to: "Is, 'ZFC + FOL → 1 + 1 = 2' true in XYZ?"

This issue touches, then, on questions of layered modality: if it is epistemically possible that a system of logic is false, but metaphysically necessary that it be true, then are we ranking epistemic modality above metaphysical modality? Hard to say! We also must, then, consider the problem of counterpossible reasoning. I am not adept at this kind of thinking, so won't pretend to be able to offer a lot of solid commentary on behalf of understanding the topic, but a sideways example is Alastair Wilson, "Counterpossible Reasoning in Physics". This article is the top result on Google for a "counterpossible reasoning" search, so hopefully it sheds a lot of light on the issue.

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From the Peano axioms it follows that there is exactly one integer which has no predecessor. We call that integer "0" so that we don't have to write "the integer that has no predecessor" all the time. Mathematicans can then prove all kinds of things about the integer named "0", for example that adding x + 0 gives the result x, for every integer x.

From the Peano axioms it follows that there is exactly one integer which is the successor of 0. We call that integer "1". And it follows that there is exactly one integer which is the successor of 1, and we call that integer 2. Later on mathematicians define an operation "+" which takes two integers and gives a result that we call the sum of the integers, and prove properties about that operations. And one property is that the successor of 0, plus the successor of 0, equals the successor of the successor of 0, or short "1 + 1 = 2".

Well, and that's it. We invented the Peano axioms which tell us about properties of integers, we gave some integers names, and we proved that with these names "1 + 1 = 2". In spoken language, we have different results: "One plus one equals two" and "Eins plus eins gleich zwei", for example. We have a special language for mathematics that makes it easier for example for a German and an English mathematician to communicate, but other cultures write simple mathematical formulas in different ways.

And all this is only true if we follow the Peano axioms. We do that because they produce lots of useful results.

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  • Note that the usual meaning of "integer" includes negative ones, so there are no integers without predecessors. What you are talking about is normally called "natural number", or sometimes "nonnegative integer". Jul 20, 2022 at 23:49
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    x + 0 = x is not a conclusion, but rather a part of the definition of addition +. a + b = b + a is a statement that you can prove using x + 0 = x. Jul 21, 2022 at 8:13
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Yes, but with the wrinkle that we can’t prove ZF is consistent. If it’s not, we would be able to prove any formal statement about sets from what would really be the contradictory premises of ZF. So proving this one would be uninteresting. And we will never be able, even in theory, to prove that any system powerful enough to represent arithmetic is both complete and consistent.

So, we can definitely know for sure that 1 + 1 = 2 is provable as a theorem in ZFC (using e.g. John von Neumann’s model of the natural numbers), but it’s somewhat arbitrary how much we care about that.

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The issue is that when people talk about "ZFC" there is a bit of abuse of meaning going on. What is commonly referred to as ZFC consists of two things, the basic propositional logic ideas and the actual ones of ZFC as it's own thing itself. I will refer to this two instances of as "ZFC with background logic" and "ZFC as theory".

Now, since whenever in practice we need to use ZFC as a theory, we need a background logic, it can be that the background logic is such that 1+1=2 false but the axioms of ZFC as a theory in itself is still consistent.

Actually, if we were to remove the background logic which allows us to make deduction (this would lead to unusuable ZFC btw), there is absolutely no contradiction as such.

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First you need to define what it means for something to be true. The simple fact is, that you are always making implicit assumptions, like the meaning of the words you use. You cannot formally define all the words that you use, because you always need to rely on words to define other words. Even in the last sentence, you may ask: And what does it mean to "formally define" something? And I'll ask back: And what does it mean to "mean" something? You see, we are getting nowhere.

So, for practical purposes, you need to rely on having a common understanding of some terms with your audience. Otherwise you would not be able to communicate at all.

To make matters worse, different people use different terms as the foundations for their definitions. Leading to the confusing fact that some people use A to define B, while others use B to define A, and you rightly complain that this yields a circular definition of both A and B. (Anecdote: My math professor used functions to define tupels, and tupels to define functions... Nevertheless, I've learned a lot from him.)

So, you cannot do without implicit assumptions. And when you ask whether 1 + 1 = 2 is an absolute truth, you need to define the meaning of "absolute truth", "1", "+", "=" and "2". Only once you have pinned down the exact meaning of these terms will you be able to answer the question. But, be aware that you will have made other implicit assumptions on the terms you used to define the terms above. At some point, you simply need to give up.

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