Gödel's results essentially conclude that there are True but Unprovable statements in arithmetic.

My thoughts are as follows:

Axioms form the foundation of mathematics -because we need to assume few things to get started. That is, in emptiness, where everything is possible, we assume few things, and then we create (or discover) new things -taking care that things stay consistent i.e. do not contradict each other.

What I wish to ask is: aren't the unprovable yet true statements (from Gödel's results) a consequence of the foundational assumptions (axioms) in the first place?

When we assumed the axioms, we did not prove them, but took them for granted. If a consistent system has been built on top of these axioms, only thing doubtful which remains are the axioms themselves. Anything then which, within theory, cannot be proved, but is true, should arise because of these axioms themselves. The hypothesis: Godel's statements are unprovable within the theory because they are the manifestations of the axioms themselves. They are true (within the theory) because they are a result of the axioms and the consistent theory developed upon them, but we find them to be unprovable. The source of this unprovability can only be attributed to the axioms we assumed, but did not prove. In effect, axioms, through the developed theory, only get translated to produce Gödel's statements.

These axioms, upon their assumption, should automatically set some constraints (because they impose constraints on what is allowed) in the empty space (which allows everything) -and these constraints can be said to be reflected through the existence of Gödel's unprovable (like our assumed axioms) but true statements (because our system, by incorporating axioms, assumes them to be true).

It is then also easy to see that mathematics is entirely a creation of mind, and not a platonic substance (as Gödel claimed).

The question therefore becomes:

If we could, hypothetically, prove all axioms perfectly, will there still exist Gödel's true but unprovable statements?

closed as unclear what you're asking by Eliran, Frank Hubeny, Mark Andrews, Nick, Conifold Apr 1 at 3:22

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  • If you have a text you are reading related to this it might help focus the question. – Frank Hubeny Mar 31 at 23:17
  • You argument is quaint: we assumed some claims as axioms, therefore, there should also be true claims that do not follow from what we assumed? If this worked it would prove too much: elementary geometry and Boolean algebra are complete, for example, there are no unprovable true claims. The last sentence is also a non-sequitur: it is equally possible that the assumed axioms do not (or even can not) describe the "platonic substance" completely, as Gödel thought, hence the additional true claims. – Conifold Apr 1 at 3:21
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    "If we could, hypothetically, prove all axioms perfectly, will there still exist Gödel's true but unprovable statements?" In what sense "prove all" ? Starting from what ? from "emptiness" ? A Mathematical proof (but IMO this holds in general) is the deduction by logic rules of a statement from something. That siomething can be a previous proved theorem or an axiom. There is no "creation ex nihilo of truth in mathematics. – Mauro ALLEGRANZA Apr 3 at 15:13
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    "aren't the unprovable yet true statements (from Gödel's results) a consequence of the foundational assumptions (axioms) in the first place?" NO; the theorem of Peano arithmetic (the statements that are provable from the axioms) are exactly the logical consequences of the axioms. The undecidable statements found by Godel are arithmetical statement that are not logical consequence of the axioms but that are "intuitively true" (in technical terms : are true in the standard model of natural numbers but are false in some non-standard model). – Mauro ALLEGRANZA Apr 3 at 15:17
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    "Anything then which, within theory, cannot be proved, but is true, should arise because of these axioms themselves." NO; it means that the specific axioms are not "strong enough" to capture all true arithmetical facts. The fact that Godel's proof has found such a true statement that is unprovable means that the specific axioms of the theory (Peano's axioms) are not enough, but there are more than them: and G's proof is a math proof indeed. – Mauro ALLEGRANZA Apr 3 at 15:20

I think you are misunderstanding Godel's incompleteness theorem:

The incompleteness theorem says if you have a complete system (i.e., a complete "axiomatic" system), then there are true statement such that they are unprovable in that system.

So no, the true statements are not the consequence of foundational assumptions (axioms). What the theorem actually says is something else:

Given a set of consistent axioms such that you can prove statements with it, there will still be true statements such that they will not be provable with those axioms.

Suppose we have a system S such that it is complete, then there is an incomplete sentence in S, S_G such that it is unprovable. Suppose, now, that we come up with a system S_1 which includes every axiom in S plus the axiom S_G. Furthermore, this new system S_1, according incompleteness theorem, will also have an unprovable statement S_G1. From that we can devise a new system S_2, and so forth each incomplete. That said, when you say incompleteness is an implication of an axiomatic system because of its axiom, that is incorrect.

I hope that answered your question, feel free to ask for clarification.

  • The issue, in my opinion, only gets propagated, step by step. At every stage, unprovable sentences are just different manifestations of the axioms -varying only in face value, but intrinsically identical/similar. Think of them as different combinations of same set of axioms. – Ajax Apr 1 at 8:59
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    @Ajax Well, your opinion is worthless when it comes to foundations of mathematics, which is highly formal and incredibly rigorous. That said, you can opine whatever you desire, but it will not change the fact that you are simply wrong. Regards – Bertrand Wittgenstein's Ghost Apr 1 at 9:48
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    @Ajax do you want to ask: "Would we still have this problem, if we wouldn't start with axioms, which are unprovable? ...regardless of whether it is even possible to start with something else?". And then another question would be "Is mathematics entirely a creation of mind?", with possible addition: "....and not the real world written down in specific format?" ? – Marko Apr 1 at 16:25
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    @Ajax You are right, I apologize. My comment could come off as slightly condescending. Regards – Bertrand Wittgenstein's Ghost Apr 1 at 17:25
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    Lots of huffing and puffing but no answers as yet. I was hoping to read one since I find the OPs question a good one and don't know the answer. It's an idea I haven't come across before. . – PeterJ Apr 2 at 11:37

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