Questions tagged [metamathematics]

the mathematical discipline that applies mathematical methods to the study of mathematical theories themselves.

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7
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2answers
259 views

Equivalences of $n$-categories

This question is an extension of my previous question last year (see [2020]) in which I asked about the (consensus of a) definition of a weak $n$-category. Here are some background: while strict $n$-...
32
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3answers
4k views

Top-down mathematics, or "Where it all begins"

Sorry if this is off-topic. It was my attempt to take a top-down approach to mathematics. Being an inexperienced undergraduate (so please take my writing here lightly), I've been presented with ZFC as ...
12
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1answer
674 views

So after all, what is this thing about topos theory and non-binary truth?

Disclaimer. The question below is necessarily vague. I understand neither the subject matter topos theory nor the object about which my question is (the construction of a fractional / non-binary / ...
13
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3answers
767 views

Is there a semantics for intuitionistic logic that is meta-theoretically "self-hosting"?

One can study the standard semantics of classical propositional logic within classical logic set theory, so we can say that the semantics of classical logic is meta-theoretically "self-hosting&...
3
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1answer
247 views

Is there an equivalent of the incompleteness theorems/halting problem in category theory?

Taking the doctrine of computational trinitarianism ( https://ncatlab.org/nlab/show/computational+trinitarianism ), if one understands the incompleteness theorems as the "logic" version, and ...
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3answers
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What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that ...
4
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0answers
232 views

How can I prove that primitive recursion “preserves” representability in Peano Arithmetic?

I'm working on my thesis about Gödel's Incompleteness Theorems, and at some point I need to prove that the $\textsf{PA}$ system is able to represent all the recursive functions. By recursive function ...
7
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1answer
353 views

Natural $\Pi_1$ sentence independent of PA

Order invariant graphs and finite incompleteness by Harvey Friedman gives an example of a combinatorial/non-metamathematical $\Pi_1$ sentence that is independent of ZFC. Is there a simpler example of ...
-4
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1answer
177 views

Is there a procedure to derive models from axiomatic systems? [closed]

Is there a systematic procedure to construct a model of an axiomatic system from the system itself? For example given the abstract postulates of a ring we can show that the integers satisfies them ...
2
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1answer
245 views

Possible to prove a lemma from Godel's completeness theorem in intuitionistic logic?

I am trying to determine whether a particular theorem used in the course of Godel’s (1930) proof of the completeness of predicate logic could be proven in an intuitionistic metatheory. Theorem VI (p....
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534 views

Is there any theorem achieving Conway's "Mathematician's Liberation Movement"

John Conway on in the appendix to part zero of ONAG describes a "Mathematician's Liberation Movement". The goal would be to give mathematicians the freedom to create mathematical theories with the ...
2
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1answer
283 views

ZFC ability to express truth and $\omega$ - consistency

Some theories can lie about their own consistency (for example, if $PA$ is consistent then the theory $PA + \lnot CON(PA)$ is consistent, although it proves its own inconsistency). Now working with ...
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0answers
76 views

Questions in proof theory (PRA-provability of EA-theorems, Girards book from '87)

I've been working through a textbook, often encountering difficulties with the exercises. On mathstackexchange, with most of them I haven't arrived yet at a satisfactory solution. As I understand, ...
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0answers
115 views

Are the HBL derivability conditions necessary for Gödel's incompleteness theorems? (For Löb's theorem?)

I've been working through a textbook, often encountering difficulties with the exercises. On mathstackexchange, with most of them I haven't arrived yet at a satisfactory solution. As I understand, ...
1
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0answers
92 views

n-consistency - provability/truth of $\Sigma^0_n$ and $\Pi^0_{n+1}$ -formulas; n-consistent extensions, etc

I've been working through a textbook, often encountering difficulties with the exercises. On mathstackexchange, with most of them I haven't arrived yet at a satisfactory solution. As I understand, ...
18
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1answer
317 views

Proof as a Σ₁ approximation to truth: what about higher degrees?

Let us posit that the goal of mathematics is to study mathematical truths, and let us stick to arithmetic for simplicity: so let $\mathscr{T} \subseteq \mathbb{N}$ be the set of (Gödel codes of) true ...
3
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3answers
685 views

How can you formalize the metamathematics conventionally used to state Godel’s theorem?

Gödel’s incompleteness theorem states that for any sufficiently strong formal system $T$ there exists a statement $G$ such that if $T$ is consistent, then $G$ is true but not provable in $T$. But I’m ...
3
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1answer
254 views

Unorthodox constructive reasoning: The Kleene Getaway

KG (the Kleene Getaway) is the name I improvised (in my answer to a question on MO on constructive Perron-Frobenius) for a constructive principle which enables the direct constructive use of a ...
3
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0answers
258 views

Can we prove the epsilon theorems without the axiom of choice?

Hilbert's epsilon $\epsilon$ is a quantifier. It follows the rule that if $\exists x. p(x)$, for some predicate $p$, we can infer $p(\epsilon x. p(x))$. Semantically, it represents picking some ...
7
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2answers
545 views

Is every true statement independent of $PA$ equivalent to some consistency statement?

Most true statements independent of PA that I know of is equivalent to some consistency statement. For example Con(PA), Con(PA + Con(PA)), Con(PA + Con(PA) + Con (PA + Con(PA)), $\dots$ Goodstein's ...
14
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1answer
1k views

What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem? In particular, what is known about the arithmetic systems $PA + \...
4
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2answers
446 views

Experiments physically performable in a finite amount of time whose results are independent of ZFC [closed]

In On independence and large cardinal strength of physical statements we see that their are physical statements which are independent of ZFC, and even strong cardinal axioms. There were many answers, ...
2
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1answer
126 views

Internal operations on uncomputable functions

Is there know set of operations for which uncomputable functions are, let's name it down-unclosed? I mean a set of operations which takes two ( or more) uncomputable functions and return computable ...
4
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2answers
676 views

What things does ZFC not know if it knows?

The statement "ZFC $\vdash 0=1$" is independent of ZFC due to Goedel's second incompleteness theorem. That got me wondering, for what other statements $\phi$ is "ZFC $\vdash \phi$" independent of ZFC? ...
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4answers
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The enigmatic complexity of number theory

One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...
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1answer
895 views

Question about the validity of Michael Grossman, Robert Katz and Jane Grossman Non-Newtonian / Meta / Multiplicative Calculus [closed]

I am no mathematician by far, but I have studied diff. and integral calc and beyond, in my undergrad years. I recently came upon this book: "The First Systems of Weighted Differential and ...
4
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1answer
362 views

Can adding Hilbert's epsilon to a theory, and then expanding that theories axiom schema to include the new language, cause it to become inconsistent?

Epsilon Calculus is a formalism developed by Hilbert adding his $\epsilon$ operator to predicate logic. $\epsilon x. A(x)$ is a term such that $\exists x.A(x) \implies A(\epsilon x.A(x))$. In can ...
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2answers
690 views

Can you remove all the extra arithmetic from ZFC (or other theories)?

Let $\mathbb{N}$ be the standard model of the natural numbers. For any statement in the language of arithmetic, we can translate into a statement in the language of set theory by asking if it is true ...
6
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1answer
315 views

What drawbacks are there to using NF(U) for category theory?

In category theory, you often run into what is known as "size" issues. That is, you run into the issue that the categories you try to define are too "big" to be sets, and so you need to use classes or ...
19
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1answer
2k views

Can you have a type theory where there is type of all types?

Normally in a type theory, you can not have a type of all types, due to Girad's paradox. This is somewhat similar to how in set theory, you cannot have a set of all sets. Therefore, usually you just ...
3
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1answer
292 views

Can you formulate a theory stating that a truth predicate does not exist for first order set theory?

A truth predicate for first order set theory would allow you to determine the truth of statements in first order set theory. A definition is given here. My question is, can you formulate a statement ...
3
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231 views

A theory which denies the existence of a truth predicate

Is there a paper or other reference that explores the implications of a theory that denying the existence of a truth predicate in its own language (perhaps based on ZFC or PA)? I know that a theory ...
2
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1answer
542 views

Is ZFC plus a truth predicate capable of variable substitution consistent?

Let us define a truth predicate that allows one to substitute in variables. For example, for a 3-ary predicate $p$ and sets $A$, $B$, and $C$, then $\text{true}(\ulcorner p \urcorner, A, B, C)$ (where ...
5
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2answers
437 views

What two ordinals are these (based on definable ordinals)?

Let $D$ be the set of definable ordinals. An ordinal s is definable if there is a predicate $p$ (in the language of (first-order) set theory), such that $p(x) \iff x=s$ for all $x$. This is definitely ...
7
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3answers
485 views

On structures that are not submitted to compatibility conditions

In mathematics, it seems that 100 % of the time, we deal with objects that have a number of different structures that are "compatible" with each other. For example, a Lie group is a manifold and a ...
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0answers
108 views

Can $GPK^{+}_{\infty}$ + $AC_{WF}$ prove "$ZFC$ proves that the class of ordinals is not weakly compact for definable classes?

Consider the topological set theory $GPK^{+}_{\infty}$ +$AC_{WF}$, where $GPK^{+}_{\infty}$ is defined as follows (this from Olivier Esser's papers, "An Interpretation of the Zermelo-Fraenkel Set ...
7
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3answers
284 views

Automatic transfer of pointwise metric computations to bundle computations

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\deriv}[2]{\frac{d#1}{d#2}}$ $\newcommand{\sAverage}[1]{\langle#1\rangle} $ $\newcommand{\IP}[2]{\sAverage{#1,#2}}$ $\...
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0answers
265 views

the strength of saying "each sentence of true arithmetic has a recursive proof"

Let $PA_{\omega}$ be just like $PA$ except that $PA_{\omega}$-proofs can use any number of applications of the recursive $\omega$-rule. The recursive $\omega$-rule allows the following: For each ...
19
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2answers
2k views

Which kind of foundation are mathematicians using when proving metatheorems?

Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. For every two objects/sets $a,b$ one can ask whether $a=b$ or not....
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1answer
808 views

Is there any set theory $T$ such that $T$ plus true arithmetic is complete with respect to statements in set theory?

Is there an effective set theory $T$ such that $T + $$TA$ is consistient and complete. It should at least prove all theorems of $ZF$ true, so that it is a "standard" set theory. In particular, the ...
2
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0answers
349 views

The theory of frames and locales as elementary topology [closed]

In What is elementary geometry? (pdf) Alfred Tarski defined elementary geometry to be that part of Euclidean geometry which can be formulated and established without the help of any set-...
7
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0answers
237 views

Gleason’s Theorem for non-separable Hilbert spaces: On a theorem of Solovay

Gleason’s theorem plays an important role in the foundations of quantum mechanics. On the positive side it demonstrates how the probabilistic structure of quantum theory follows from its logical ...
5
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1answer
148 views

Is it possible to prove concentration bounds from optional stopping theorem?

It is known that the optional stopping theorem from martingale theory is a very powerful theorem in probability theory in statistics. I have heard of a probability course at Stanford where ...
15
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3answers
4k views

Is there a compendium of the consistency strength between the most important formal theories?

Preliminar Notions: A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...
12
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2answers
1k views

Existential statement without witness

Are there existential theorems of ZFC, or PA say, with no witnesses? Ie does there exist a formula $\phi$ such that ZFC $\vdash\exists x \phi(x)$, but for all numerals $\underline{n}$, ZFC $\nvdash \...
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2answers
378 views

Rationale behind an requirement on Turing machines

Hopcroft and Ullman's definition of a Turing machine seems to be standard. This definition defines a Turing machine to be a 7-tupel $T = \langle Q,\Gamma,b,\Sigma,\delta,q_0,F \rangle$ obeying some ...
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1answer
496 views

Is there any current development of a first order formalization of metamathematics?

I hope that this post isn't off topic, but I already asked math.stackexchange about first order formalizations of first order logic. There are provability logics and extensions in modal logic's that ...
9
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2answers
3k views

Meta$^{n{-}th}$ mathematics [duplicate]

Metamathematics has a reasonably clear connotation, enough to have a Wikipedia page, with Gödel, Tarski, and Turing playing leading roles; Kleene's book (Introduction to Metamathematics (Amazon link));...
12
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2answers
783 views

How necessary is Godel's Condensation Lemma

It seems that the Godel's Condensation Lemma is typically used to show that certain constructible sets will appear by some stage of the construction of $L$. For example in the proof that CH holds in $...
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1answer
322 views

Nontrivial, partially uncomputable function

is there any example of function which is computable on some set and uncomputable on other set? That is for example function f(n) which is computable on some (finite, or for example for even numbers) ...