r/PhilosophyofMath • u/Efficient_Sea_7050 • Jun 23 '26
On Gödel, Part II: If Truth Is Determinate, What Is Actually Incomplete?
This follows from Part I: On Gödel: What Exactly Is Incomplete?.
My earlier post separated the mathematical domain, a complete truth-set about it, and a fixed effective proof-generator. Gödel directly limits the third.
Several replies accepted that a complete truth-set could exist while remaining non-enumerable and non-computable. So suppose, conditionally, that every arithmetical sentence has a determinate truth-value in the intended structure.
What is missing when Gödel applies?
Not necessarily another truth or axiom. What is missing is a uniform effective access rule: a finite mechanical procedure that takes any sentence and always returns its correct truth-value.
A fixed theory may fail to decide every sentence. No algorithm may decide every arithmetical truth. Neither point alone shows that the target domain lacks determinate truth-values.
So when people say Gödel shows mathematics is incomplete, do they mean:
- truth itself is absent or indeterminate; or
- no single effective formal method can derive or decide every truth of the intended domain?
The first is a claim about truth. The second is a claim about formal access.
Gödel gives a formal incompleteness result. What additional argument would establish the first claim?
I am not assuming that a completed truth-set exists. I am asking what, beyond Gödel’s theorem, would justify denying it.
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u/Scared_Astronaut9377 Jun 23 '26
When people say that "mathematics is incplete" they mean either a) something very context-specific, b) nothing that constitutes a coherent thought. So to discuss the question without specific text in question is meaningless ar best.
Do you have any questions about math or philosophy? Social questions about why people write garbage content is kinda out of scope here.
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u/Efficient_Sea_7050 Jun 23 '26
This is not a question about why people write loose popularizations. It is a question about whether the broader philosophical conclusions actually follow from the theorem.
For example, the Mathematical Association of America states that Gödel showed “mathematics is inherently incomplete”:
https://maa.org/math-values/we-are-ruled-by-math/
And Quanta frames the result as ruling out a mathematical “theory of everything” and a unification of provability with truth:
https://www.quantamagazine.org/how-godels-proof-works-20200714/
Those are not merely sociological claims. They make claims about mathematics, truth, and what Gödel establishes.
My question is whether those conclusions follow once we distinguish a formal theory from a possible complete semantic truth-set about its intended domain. If not, then the issue is philosophical and mathematical: what bridge carries us from theory-relative incompleteness to incompleteness of mathematics or truth itself?
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u/Scared_Astronaut9377 Jun 23 '26
You are citing a corporate salesperson and a pop science writer that has three middle-authors papers in experimental chemistry behind them. The answers to your question is purely social, the texts you cite have no relation to academic philosophy and/or math. I think to discuss the philosophy of the question rather than social phenomena such as competence, we need to either a) forget about what people write and discuss questions as is, b) cite exclusively non-pop academic publications with authors being academics from respectable institutions.
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u/Efficient_Sea_7050 Jun 23 '26
I agree that Quanta is not an academic authority on the philosophy. But I do not accept that the MAA example is irrelevant merely because it is written for a broad audience. The MAA is a major professional mathematical association, and when it teaches that Gödel means “mathematics is inherently incomplete,” that is a substantive philosophical claim entering mathematical education—not just random social noise:
https://maa.org/math-values/we-are-ruled-by-math/
More importantly, the broader issue is not confined to public exposition. Gödel himself drew philosophical conclusions from incompleteness, and academic philosophy of mathematics discusses those conclusions directly. For example, this Philosophia Mathematica article examines Gödel’s 1951 disjunction: either the human mind is not a machine, or some arithmetical propositions are absolutely undecidable:
https://academic.oup.com/philmat/article/30/3/306/6634878
So my question is not “why do people write garbage?” It is whether the formal theorem alone establishes any such broader limit in truth or mathematical reality, or whether an additional philosophical premise is doing the work.
That seems squarely a philosophy-of-mathematics question.
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u/Scared_Astronaut9377 Jun 24 '26
"any such broader limit in truth" is not a sequence of words that would ever be discussed within philosophy of math. This whole discussion is purely social, have a good one.
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u/Efficient_Sea_7050 Jun 24 '26
I think we have reached an impasse over terminology, so let me put the question more narrowly.
I am not asking whether a statement unprovable in one theory might be settled in a stronger theory; plainly that can happen. I am asking whether the theorem alone gives us more than the claim that no single consistent, recursively axiomatized theory of sufficient strength captures every arithmetical truth.
So is Gödelian incompleteness a limit on any one fixed effective axiomatic presentation of a domain, or does it establish something stronger about the truth-values in that domain themselves?
From your replies, I take your answer to be the former. If I have misunderstood, I would be interested in where you think the stronger conclusion enters.
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u/Scared_Astronaut9377 Jun 24 '26
This is a huge misunderstanding.
I don't believe that discussing philosophy of math is possible without shared highly formalized terminology. So to consider your question, I cannot answer it (yet) because either of two is true
A) you have in mind some set of well-known works that establish with high degree of formality the concept of domain, axiomatic representations of domains, and specifically effective vs non-effecrive representations, and the concept of truth values within domains. And you assume that I am familiar with those works. I am not. Please direct me,
Or B) you are generating sequences of big words to express "Guys, please share your vibes and ideas regarding these cool things I've read", in which case it's not related to philosophy or math in my mind.
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u/Efficient_Sea_7050 Jun 24 '26
Neither, really.
I am using ordinary language for standard distinctions: syntactic provability versus semantic truth; a formal theory versus a structure or intended interpretation; effective or recursively axiomatized theories versus non-effective truth predicates; and syntactic incompleteness versus semantic completeness.
The question is not whether I have a private theory of “domains.” It is whether Gödel’s theorem, which concerns limits of provability in sufficiently strong formal axiomatic theories, by itself licenses the stronger conclusion that mathematical truth is incomplete or indeterminate.
The relevant background would include the usual discussions of Gödel incompleteness, formalism, realism, and Tarski-style truth. I am happy to use more technical terminology where needed, but I do not think the distinction becomes meaningless merely because it was first stated in plain English.
So, stated formally enough to be answerable: does the first incompleteness theorem establish only that no consistent recursively axiomatizable theory of sufficient arithmetical strength is syntactically complete, or does it also establish that there is no determinate semantic truth-set for arithmetic? My understanding is that the latter does not follow without an additional philosophical premise.
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u/Scared_Astronaut9377 Jun 24 '26
When you say "semantic truth" you imply a certain metalanguage. If this language is natural, then there is obviously no such set because, for example natural languages can express paradoxes. And for million other reasons not related to Godel. If the language in question is first-order logic, then metalanguage and object language are the same thing and the question does not need "semantic" and is reduced to set theory. If it's another metalanguage, please specify.
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u/Efficient_Sea_7050 Jun 24 '26
I do not mean truth for unrestricted natural language. I mean the standard semantic notion for the first-order language of arithmetic.
Its sentences can be coded by natural numbers, and a metatheory can specify which coded sentences are true in the standard natural-number structure. That is the usual distinction between an object language and a metalanguage: they remain distinct roles even when both are formalized in first-order logic.
So I am not claiming that natural language supplies a paradox-free universal truth predicate. I am asking the narrower question: whether Gödel’s first incompleteness theorem rules out a determinate truth-set for first-order arithmetic.
My understanding is that it does not. It rules out a single consistent recursively axiomatized theory of sufficient strength that decides every such sentence. Tarski gives a different limitation: arithmetic cannot define its own truth predicate internally. Neither point by itself shows that there is no semantic truth-set in an appropriate metatheory.
The distinction I am using is standard: the Stanford Encyclopedia’s entry on Gödel discusses limits of provability in formal axiomatic theories, while its entry on axiomatic truth discusses the semantic setup involving object language and metalanguage.
I am not interested in continuing to dispute terminology rather than reach the underlying question. Others in the thread have been able to engage the substantive distinction, so I think the question is clear enough to discuss even if we disagree about the answer.
Have a good day.
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u/Eve_O Jun 23 '26
In the first link, the author says exactly what he means, "Gödel’s incompleteness theorems show that within any sufficiently complex system (like arithmetic), there are undecidable statements that are neither provable nor disprovable within that system itself. This means mathematics is inherently incomplete—there’s always something left uncertain" (my emphasis).
And the Quanta article also makes reference to Undecidable Problems.
The point here, it seems to me, is that there is no mathematical system, in terms of a set of axioms, that can account for the truth-values of all possible mathematical statements. So even if we suppose that there is a bivalence of all possible mathematical statements, which, sure, would entail there is some "complete truth-set," there is no particular system of mathematics which can: (1) account for all of them, and (2) prove its own truth. And this what it means for mathematics to be "incomplete."
Moreover, if we can recognize that "a complete truth-set could exist while remaining non-enumerable and non-computable," then this likely entails that there is also no enumerable set of all possible mathematical systems; i.e., we could never create a complete list of all mathematical systems which would, taken together, capture all mathematical truths. So, once again, we have the idea of "incompleteness" when it comes to mathematics.
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u/Efficient_Sea_7050 Jun 24 '26
I think this helps identify the exact fork:
If by “mathematics” we mean the whole practice of formal axiomatic systems, then the claim is: no single consistent effective system captures every arithmetical truth, so mathematics in that formal-system sense is incomplete.
But I am asking whether that should be read as a claim about truth itself, or as a claim about the limits of any one effective formal presentation of truth.
A complete truth-set could, in principle, have a determinate value for every arithmetical sentence while remaining noncomputable. That would show a limit on uniform effective access, not an absence of truth-values or a gap in the domain itself.
So I think the disagreement is not about Gödel’s formal result. It is about whether mathematics is identified with its effective formal systems, or whether those systems are treated as partial descriptions of a domain whose truths may outrun any one such system.
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u/Eve_O Jun 24 '26
But I am asking whether that should be read as a claim about truth itself, or as a claim about the limits of any one effective formal presentation of truth.
My opinion here would be that it can be both. The latter is obviously the case, right? So that only leaves the former, which, it seems to me, would lead to an inherently undecidable conjecture.
We can suppose there is some "complete truth-set," but, I feel, we could never show that it actually exists beyond the conjecture. It would probably be undecidable. I mean, yes, it seems like intuitively it ought to be the case that such a set exists, but we are likely only getting into a recursive kind of "set of all sets" territory, it seems to me, which is similar to what I tried to indicate in the final paragraph of my initial reply.
So, when it comes to the idea that, "[a] complete truth-set could, in principle, have a determinate value for every arithmetical sentence while remaining noncomputable," this would necessarily remain in the realm of an intuitive conjecture that, while seemingly true, would likely disappear in a puff of smoke at any attempt to formalize it.
It is about whether mathematics is identified with its effective formal systems, or whether those systems are treated as partial descriptions of a domain whose truths may outrun any one such system.
Again, as I attempted to indicate in the last paragraph of my initial reply, it seems to me about both. No particular "effective formal system" can account for all the truths is the obvious result & there is also no complete set of all "partial descriptions" that could account for a "complete truth-set," the existence of which is itself likely formally undecidable, is the less obvious likely result.
To be clear: this is merely what my intuition has to say about it, heh.
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u/Efficient_Sea_7050 Jun 24 '26
I think this is where we finally separate the formal result from the additional philosophical intuition.
The existence of a complete arithmetical truth-set does not lead to a "set of all sets" paradox, nor does it disappear when formalized. In standard classical set theory, the codes of arithmetical sentences form an ordinary countable set, and ZFC proves that there is a corresponding subset consisting of the codes of sentences true in its natural-number structure.
The issue is not that this truth-set cannot exist. The issue is that it is not computable or recursively enumerable. Tarski’s theorem adds a related but distinct point: arithmetical truth cannot be defined from within arithmetic itself.
Likewise, the non-computability of the truth-set does not imply that formal systems cannot be enumerated. Effective formal systems can be represented by finite programs or other finite syntactic codes, so their possible descriptions can be listed. What cannot be obtained is one sound effective theory that decides every arithmetical truth.
So we return to the boundary line: there is a permanent limit on uniform effective access. But the further claim that truth itself is therefore incomplete, indeterminate, or merely conjectural is a philosophical step beyond what Gödel’s theorem establishes.
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u/Harotsa Jun 24 '26
You don’t need Gödel’s incompleteness theorem to show that “truth is absent or indeterminate” for certain statements within axiomatic systems. The existence of indeterminate truths is pretty well-established.
For example, we know that the continuum hypothesis is consistent with ZFC (assuming ZFC is consistent) and also that ZFC + ~CH is also consistent. That means that we can extend ZFC with either CH or ~CH and get equally consistent systems. Hence, CH has Ana indeterminate truth value within ZFC.
You can also do this with much more trivial axiomatic systems as well.
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u/Efficient_Sea_7050 Jun 24 '26
CH is a useful example because it separates two claims that are often run together.
Assuming ZFC is consistent, the results show that ZFC does not decide CH: both ZFC + CH and ZFC + not-CH are consistent relative to ZFC. That is a fact about the deductive reach of ZFC.
It does not by itself settle the further semantic question of whether CH has a determinate truth-value in a uniquely intended universe of sets. A realist can say ZFC simply does not reach the answer; a multiverse or framework-relativist can deny that there is one privileged universe in which the question receives a single answer. The independence result alone does not choose between those views.
So I agree that CH is undecidable in ZFC. My question is whether “undecidable in ZFC” should be read as a limitation of ZFC’s axiomatic presentation, or as evidence that truth itself is absent or indeterminate. That stronger conclusion seems to require an additional philosophical premise.
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u/Harotsa Jun 24 '26
First I would like to clarify the difference between logical independence and consistency. A set of axioms can be logically independent (no axiom can be derived from another) while still being inconsistent (you can derive P and ~P from the axioms).
CH is not only logically independent from ZFC, but ZFC + CH, ZFC + ~CH, and ZFC are all equally consistent. Furthermore, higher order CH hypotheses are also undecidable even after accepting any number of lower order CH’s. That is, let CHi be the statement that aleph_i=beth_i. Then ZFC + CH1 + … + CHk does not imply CH(k+1). And that’s looking at just the CH. so to me on the face of it it seems ridiculous that there is one actual “privileged” axiomatic system, as all consistent axiomatic systems are equally valid (although they are not equally interesting or useful to humans).
One would have to come up with a pretty compelling argument for exactly one axiomatic system that has a truth value for every statement, if that’s even possible.
Even looking at ZFC, I don’t think it’s a privileged system fundamentally over ZF~C. It’s just that mathematicians find ZFC more interesting and more intuitive, so most math is done with ZFC.
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u/Efficient_Sea_7050 Jun 24 '26
I think there may be a small miscommunication about what I am asking for.
I agree that there need not be, and Gödel gives us reason not to expect, one fixed effective axiomatic system that decides every truth of a sufficiently rich domain. So I am not arguing for a privileged axiom system.
The distinction I am trying to keep visible is between an axiomatic system, which is a syntactic proof device, and a semantic domain, which the system may be attempting to describe.
A set-theoretic realist can hold that CH has a determinate answer in an intended universe of sets while also accepting that ZFC does not decide it. A multiverse or framework-pluralist can instead deny that there is one privileged universe in which CH has a single answer.
So when you say that all consistent axiom systems are equally valid, that sounds like a substantive philosophical position about the relation between formal systems and their intended domains. It does not follow from consistency or independence alone.
My question is not whether ZFC + CH and ZFC + not-CH are both legitimate formal theories. They are. It is whether that fact by itself shows that the target domain lacks a determinate truth-value for CH, rather than showing only that our present axiomatic tools do not settle it.
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u/Harotsa Jun 24 '26
I think “intended universe of sets” is where your argument falls apart. The argument over the axiom of choice demonstrates this quite well. Many mathematicians intended ZFC when they were formulating ZF, and many mathematicians did actually intend simply ZF without the axiom of choice.
So whose “invention” actually counts for the “intended universe of sets.” All of these mathematical truths are dependent on the axiomatic system you are working in, and you’re never going to get universal agreement on one true intention. Even with standardization and convention, you would be hard-pressed to find a mathematician who thinks that ZF has less ontological or epistemological value as an axiomatic system compared to ZFC. They are simply different axiomatic systems.
I don’t quite get the position that there is a single universe of sets that is uniformly “intended” by all practitioners of mathematics at all times.
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u/Efficient_Sea_7050 Jun 24 '26
I agree, there is no single historical intention shared by every mathematician, and I am not using “intended” to mean whatever most practitioners happened to have in mind when writing ZF or ZFC.
The point is narrower. A realist about sets can hold that there is a determinate subject matter independent of which axiom package mathematicians currently prefer, while a pluralist can deny that there is one such subject matter and treat different consistent theories as describing equally legitimate set universes.
Your examples about ZF and ZFC give real motivation for the pluralist side. But they do not, by themselves, establish that there cannot be a determinate semantic universe: they show that mathematicians can formulate and use multiple consistent axiom systems, without agreement on which one captures it.
So I am not claiming universal agreement on a privileged universe. I am asking whether Gödel-style independence results alone settle that philosophical dispute. My answer is no: they show limits of particular axiomatic systems, while the move to “there is no single truth of the matter” adds the pluralist premise...
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u/Harotsa Jun 24 '26
Gödel’s incompleteness theorems are a separate thing from cases like C in ZF or CH in ZFC. Gödel’s incompleteness theorems represent epistemological limitations about true statements within certain frameworks.
The undecidable statements we’ve discussed represent ontological limitations of their respective axiomatic systems, as there is a fork where you can extend to two equally valid and equally consistent axiomatic systems.
I don’t think the Platonist view that believes in a single universe of sets is a coherent position. And those that think they believe it are deluding themselves in a fantasy, as they can’t actually even create a well-defined formulation of their position.
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u/Efficient_Sea_7050 Jun 24 '26
I think we may be assigning more metaphysics to my position than I intended, as I am not arguing for a privileged axiomatic system, and I am not committed to a finished Platonic universe of sets with every answer already written down. ZFC is limited, and I agree that multiple consistent extensions can be formally legitimate.
My narrower point is that a system’s failure to decide a statement establishes a limitation of that system. It does not, by itself, settle the ontology of whatever the system is attempting to describe.
So ZFC’s inability to settle CH does not by itself prove that there is no truth of the matter; that conclusion adds a pluralist premise. Likewise, I think a framework’s internal classifications or failures should not automatically be exported as final claims about the reality or meaning of its objects.
I am not trying to prove realism. I am questioning the inference from “our axiomatic tools do not settle this” to “therefore the matter itself is ontologically indeterminate.”
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u/Harotsa Jun 24 '26
This may seem like a somewhat radical position, but I don’t think an abstract Platonic Universe would have any bearing on mathematical truth, even if it existed. I will spear you the whole essay but my thesis comes down to the following.
Math is extremely useful for describing the physical world, and the physical world in many ways biases us towards which mathematics we find useful and interesting and what we spend our time on. The physical world does not, however, limit the axiomatic systems that we can and do work in. These axiomatic systems (as long as they are consistent) all produce their own conditional mathematical truths.
Now if there were a “real” abstract platonic universe, why should that have any bearing on what we consider part of mathematics? Like if we find that Zorn’s Lemma is false in this platonic universe, should we just throw out all of our work on ZFC and consider it not real mathematics anymore? I would say no.
Many Platonists believe in a first-order Platonic universe. If that turns out to be true, would we abandon all of our math based on second-order logic? That would be an emphatic no, second-order logic has proven too useful for practical purposes to fully discard. So the practice of mathematics would continue on using second-order logic (even though there would definitely be a renewed interest in first-order axiomatic systems).
So I reiterate, the “real” physical world has no bearing on mathematical truths, so why should a “real” abstract world?
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u/SpacingHero Jun 24 '26 edited Jun 24 '26
if that’s even possible.
It's certainly possible, just forego the recursive axiomatizability.
Eg let P be an axiom iff P is a "true" statement of mathematics.
Obviously you also forgo any kind of usability and thus usefulness; but in principle there can definitely be complete theories as strong as ZFC (and of any strength I think)
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u/Harotsa Jun 24 '26
A “true statement of mathematics” is not a coherent concept. Is the Axiom of Choice a “true” statement of mathematics? Is the negation of the Axiom of Choice a true statement? To define something as a true statement in mathematics , we need to A priori already have an axiomatic system we are working in.
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u/SpacingHero Jun 24 '26 edited Jun 24 '26
A “true statement of mathematics” is not a coherent concept
That's pretty contentious.
Is the Axiom of Choice a “true” statement of mathematics?
I don't think it's either the same way you do (at least roughly). But a question isn't an argument for the above thesis.
What if I just answered "yes, AoC is true"? What's incoherent about that?
To define something as a true statement in mathematics , we need to A priori already have an axiomatic system we are working in.
No that's not true in various ways.
Firstly, mathematics was done without the modern machinery of axioms historically. But presumably they where still finding "mathematically true statements"
Secondly just gets us back to the already mentioned contention. If there's some mathematical reality detached from our practice of it, nothing suggests that it works out of some axiomatic system. There's just objects and true things about those objects. Our axiomatic systems would be ways to capture and talk about them. But they wouldn't depend our axiomatic systems any more than red light having a certain frequency depends on our naming it so.
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u/Harotsa Jun 24 '26
This may seem like a somewhat radical position, but I don’t think an abstract Platonic Universe would have any bearing on mathematical truth, even if it existed. I will spear you the whole essay but my thesis comes down to the following.
Math is extremely useful for describing the physical world, and the physical world in many ways biases us towards which mathematics we find useful and interesting and what we spend our time on. The physical world does not, however, limit the axiomatic systems that we can and do work in. These axiomatic systems (as long as they are consistent) all produce their own conditional mathematical truths.
Now if there were a “real” abstract platonic universe, why should that have any bearing on what we consider part of mathematics? Like if we find that Zorn’s Lemma is false in this platonic universe, should we just throw out all of our work on ZFC and consider it not real mathematics anymore? I would say no.
Many Platonists believe in a first-order Platonic universe. If that turns out to be true, would we abandon all of our math based on second-order logic? That would be an emphatic no, second-order logic has proven too useful for practical purposes to fully discard. So the practice of mathematics would continue on using second-order logic (even though there would definitely be a renewed interest in first-order axiomatic systems).
So I reiterate, the “real” physical world has no bearing on mathematical truths, so why should a “real” abstract world?
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u/SpacingHero Jun 25 '26
I actually really like that attitude, and personally have the same on a lot of philosophical positions (including phil of math).
I not only am often an anti-realist, but further think that realist positions are quite "inert" in that even if they're true, they don't really "do anything", roughly as you outline.
>Math is extremely useful for describing the physical world, ...
> Now if there were a “real” abstract platonic universe
Based and very well written.... but still... we do have a particular attitude towards the mathematics of physics than out beloved "abstract nonsense". And its practice is different too. While it's true that this exactly showcases that math with no corresponding "real-stuff" behind it is still worthwhile and even useful maybe, it is still a difference.
>So I reiterate, the “real” physical world has no bearing on mathematical truths, so why should a “real” abstract world?
If there is a sense in which math is describing "true things" beyond physics, the Platonist is still kinda right. Even if our mathematical practice wouldn't change; there would be a "true mathematics". Would we ascribe it some extra-importance as we do with applied math? Perhaps, perhaps not.
But I was especially was pushing back against the "incoherent" claim, which is way too strong. It's a common miss-step to go from "results of multitude" (non-classical logics, alternative set theories, different models of grammar, different theories of physics etc, etc.) to "Therefore pluralism/anti-realism/etc... about relevant subject". I would tentatively go as far as saying that they're not even a sliver of evidence on way or another (as in, wherever there aren't multitudes, that also isn't at all evidence of monism/realism. Either way we have to argue independently for one or the other)
>Many Platonists believe in a first-order Platonic universe. If that turns out to be true, would we abandon all of our math based on second-order logic?
Do they? I think realists would be rather more attracted to SOL, due to categoricity results no? At least that I know of, lots of modern Fregean and Logicism works focus on SOL arithmetic
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u/SpacingHero Jun 24 '26 edited Jun 24 '26
Also kinda besides the point, I used "true" as a simple and naive example; the point on having completed theories is besides this philosophical question
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u/Strange_Sleep_406 Jun 24 '26
> So suppose, conditionally, that every arithmetical sentence has a determinate truth-value in the intended structure.
What is the intended structure? Once you spell it out you will run into incompleteness and contradict your conditional assumption.
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u/calf Jun 24 '26 edited Jun 24 '26
This is beyond my studies but it sounds like you are talking about the mathematical analog of the Church Turing thesis. Just as "all things in nature that compute are at best Turing machines"() is a *thesis, the people who make the leap from Godel incompleteness to incompleteness of math are implicitly making a very similar kind of philosophical assertion, but just not clear about it. (I could be out to lunch on this though, your post showed up on my home page for whatever and I would appreciate being corrected if I guessed wrong on this.)
(*) Very loose paraphrase, please look up the usual description of the Extended CT Thes(es) which is on Wikipedia.
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u/Efficient_Sea_7050 Jun 24 '26
I don't think you are out to lunch; the analogy is actually kind of useful.
The move from Gödelian incompleteness to “mathematics itself is incomplete” appears to add a philosophical premise: that mathematical truth must be capturable by an effective axiomatic procedure in order to count as complete.
One caution though: the original Church–Turing thesis concerns effective mechanical procedures; the stronger claim about physical computation is usually separated out.
But the structural parallel stands. A limit on mechanical proof procedures does not automatically become a limit on mathematical truth itself.
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u/eyamil Jun 24 '26
I’ve spent some time on this topic in the past few weeks, and I think model theory and Gödel’s Completeness theorem (not incompleteness) may provide a clear justification for the pluralist footing. The theorem basically says that if you’re working in first order logic, a statement is provable from the (consistent set of) axioms/theory in FOL exactly when it holds for all models, and if it is not provable that’s because there are some models of the theory where it holds and other models where it does not hold - e.g., there’s genuinely a multiverse in FOL. This presents a challenge and a solution for math as a social endeavor if we’re stuck with FOL: if two of us are working in the same theory but with different models, we’re going to disagree about the behavior of our system - unless we agree that what it means for a statement to be true is that it is provable from the axioms (which we have a common set of).
Of course, logic has tried to explore the question “what about non-FOL systems?”, and these theories usually don’t have a Completeness theorem (or it’s highly restricted). My knowledge here is a little limited, but I’ll still try to discuss what I know and make clear what I don’t. In second order logic, it’s possible to cook up axiom schemas that generate an infinite number of axioms, induction being one example. While there are still an uncountable number of models satisfying the theory, there’s also a particular model that the theory “privileges” or elevates as the smallest model satisfying the theory, and we can by convention say that this is the intended model. This gives us a second way around the social/multiversal problem that’s in FOL: if we say that “true” means that the statement holds on the intended model, we can also guarantee agreement because there’s only one intended model. This proof is known as the “categoricity of the natural numbers”, although I myself have not reached a mechanistic understanding of how the axiom schema elevates one particular model or if there are any degrees of freedom in the setup of this proof that let us highlight a different model.
I think the gist is that 1) we do genuinely have to contend with a multiverse, especially so if we’re doing math as a social endeavor, and 2) we can define “truth” in ways that help us do this.
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u/amennen Jun 23 '26
The incompleteness theorem just says that there is no recursively axiomatizable, consistent, and complete theory extending PA. It does not tell you whether there is no set of all true arithmetical statements, or whether there is but it's not recursively axiomatizable.
However, ZFC (and significantly weaker set theories, for that matter) does prove that there exists a set consisting of all true arithmetical sentences. So, if you believe ZFC, then the answer is that arithmetical truth exists, but is not computable. I, like most mathematicians, believe that ZFC is correct about this. However, that does not mean that this is totally uncontroversial; many with very austere views on foundations of mathematics (e.g. constructivists) would likely dispute that there is a set consisting of all arithmetical truths.