Questions tagged [metamathematics]

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

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Does CZF prove there is a minimal cauchy completion of the rationals?

In IZF, we can easily prove there is a minimal cauchy complete field extending the rationals: the dedekind reals are cauchy complete, so just intersect all of its cauchy complete subfields. CZF can ...
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2 votes
0 answers
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A formal definition of a useful theorem?

Sorry if this feels a bit squishy, but I'm wondering if there is any published work trying to give a fully formal definition of the notion of a useful theorem. I mean, in mathematics we all know that ...
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2 votes
0 answers
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How restrained are we in terms of metatheories when working with higher order logics with full semantics?

When working in the realm of first order logic one can use very basic mathematical backgrounds(in reverse mathematical sense) to prove interesting things about more "structures". In what ...
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7 votes
2 answers
360 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 $n$-...
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33 votes
3 answers
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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 ...
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12 votes
1 answer
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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 / ...
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15 votes
3 answers
893 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&...
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3 votes
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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 ...
26 votes
3 answers
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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 ...
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4 votes
0 answers
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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 ...
8 votes
1 answer
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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 ...
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-4 votes
1 answer
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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 ...
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2 votes
1 answer
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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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8 votes
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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 ...
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2 votes
1 answer
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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 ...
1 vote
0 answers
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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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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, ...
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1 vote
0 answers
96 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, ...
  • 121
18 votes
1 answer
337 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 ...
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4 votes
4 answers
887 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 votes
1 answer
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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 votes
0 answers
264 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 ...
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7 votes
2 answers
584 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 ...
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15 votes
1 answer
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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 + \...
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4 votes
2 answers
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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, ...
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2 votes
1 answer
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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 ...
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4 votes
2 answers
733 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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82 votes
4 answers
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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, ...
1 vote
1 answer
968 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 votes
1 answer
401 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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1 vote
2 answers
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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 ...
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6 votes
1 answer
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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 ...
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20 votes
1 answer
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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 ...
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3 votes
1 answer
315 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 ...
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3 votes
0 answers
239 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 ...
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2 votes
1 answer
577 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 ...
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6 votes
2 answers
576 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 ...
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7 votes
3 answers
487 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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1 vote
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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 votes
3 answers
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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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4 votes
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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 ...
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19 votes
2 answers
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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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9 votes
1 answer
830 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 ...
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2 votes
0 answers
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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 votes
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255 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 votes
1 answer
158 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 ...
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16 votes
3 answers
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 votes
2 answers
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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1 vote
2 answers
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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 ...
1 vote
1 answer
536 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 ...
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