Is everything in math either an axiom or a theorem?

[jbriggs444]

everything in math is either an axiom or a theorem. There is a third possibility with undecidable statements but that's too complicated.

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

 

[SSequence]

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

This seems like a reasonable view to me.

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The way I see it simply this way. If we have a decidable theory then (assuming consistency) every statement falls exactly into one of the following two categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory

The statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].On the other hand, if we have an incomplete theory under consideration [as often happens to be the case] then (assuming consistency) every statement exactly falls into one of the following three categories:
(i) provable in the theory (or a theorem of the theory)
(ii) disproveable in the theory
(iii) independent

And once again the statements in category-(ii) that would be disproveable (in the theory) would have their negation as a theorem [theorem of the theory that is].

 

[Feynstein100]

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

Fine. It's not a duality but a trinity: Axiom, Theorem or Undecidable

 

[pinball1970]

If you slice away everything that is "too complicated", many false claims become true.

Oh, and the way I learned it, every axiom is a theorem. With a one line proof.

Yourself regarding infinitities and Cantor @PeroK generally and @fresh_42 have been valued over the years. Thank you. Fresh insights are good.

 

[EventHorizon]

Considering math as a collection of true/logically consistent statements, I see only two possibilities: either the statement is true and can be proven, which means it's a theorem. Or it's true but cannot be proven, which means it's an axiom. Is there a third possibility? Or maybe more?
I feel like we're venturing into Gödel incompleteness territory here but for this discussion, let's keep it simple. Is this duality of axiom/theorem all-encompassing or are there things that lie beyond?
It's also interesting to note that since theorems follow from axioms, it should be possible to write down all axioms and all their subsequent theorems, in a linear timelike/causality-like structure. Is this a coincidence?
And finally, where does the notion of mathematical objects fit into this? For example, you might find out all axioms and theorems related to numbers. But once you introduce the idea of vectors and tensors, which are different mathematical objects, the previous knowledge doesn't apply anymore. Because they are different objects, they will have different properties, meaning different axioms and theorems. Basically, any new mathematical object will have its own set of axioms and theorems. Which raises the question, will there always be some new mathematical object to discover i.e. there are an infinite number of them or will we eventually run out of them?

There are also things like the Navier-Stokes equation where there are no defined "solutions"

 

[Vanadium 50]

There are also things like the Navier-Stokes equation where there are no defined "solutions"

What? Are you guessing again? Please don't do that.

Of course there are solutions. Otherwise it would be fairly useless. What there are not are closed-form solutions involving only elementary functions for arbitrary initial conditions.

If you are referring to the Clay Instiute's Navier-Stokes problem, that's something rather different from what you wrote.

It's aslo kind of an off-topic necropost.

 

[renormalize]

There are also things like the Navier-Stokes equation where there are no defined "solutions"

Do you refer here to the lack of formal proofs of existence, uniqueness and smoothness of solutions of the N-S equation?

 

[Rfael]

you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D

 

[Svein]

Did we invent the numbers? Give them properties? The nomenclature is ours (Arabic, Roman, Babylonian etc) but did we invent the actual numbers?
Give them properties? Even numbers, primes?
EDIT: I know this is straying away from the OP a little.

Some number systems are better than others in expressing mathemathical conjectures. The Peano axioms are meaningless in roman numerals - leaving aside the fact that 0 does not exist, the notion of a "successor" is not trivial (what is the successor of VIII?)

 

[jbriggs444]

Some number systems are better than others in expressing mathemathical conjectures. The Peano axioms are meaningless in roman numerals - leaving aside the fact that 0 does not exist, the notion of a "successor" is not trivial (what is the successor of VIII?)

I suppose that we could shift our numeral system to a Gray code. Though I doubt it would make constructing the reals from the Peano axioms any easier.

 

[FactChecker]

I've been meaning to ask about that. It seems to me that what Gödel actually discovered is that self-reference is different kind of thing that isn't compatible with normal everyday logic i.e. the pattern of true/false doesn't necessarily apply to self-referential statements. I have yet to see an example of it in a non-self-referential context i.e. a statement that isn't self-referential and not true/false. It seems that as long as you stay away from self-reference, you should be fine.

I think it is important to distinguish between the number of counterexamples that are easily proven to be counterexamples versus the number of counterexamples that exist. There are $\aleph_1$ transcendental numbers, but far fewer proven ones.

 

[Henryk]

I'm certainly not a mathematician, can you give an example of something known to be true but unproven? In my mind, axioms are "given" -- assumed to be true. But they may not "really" be true. Things like flat space.

There is no such thing as not "really" true in mathematics. Maybe the general theory states that the physical space is not actually flat, but that doesn't mean we can't construct a true flat space in mathematics.

 

[Feynstein100]

you have 'axioms' 'theorems' 'lemmas' and 'corollaries' :D

Just when I thought I was done, they pull me back in

 

[Hornbein]

Some systems are decidable. Every statement can be proved or refuted within the system. Other systems are not. See "Hilbert's tenth problem". Some limited systems of Diophantine equations are decidable, others are not. It matters how many unknowns one allows and other things. Julia Robinson did much work with this.

There can be a system with an infinity of axioms but this unattractive.