# Questions tagged [metamathematics]

The metamathematics tag has no usage guidance.

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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 ...

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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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### 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, ...

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### 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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### 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 ...

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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 ...

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### 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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### 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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### Indirect attempts at showing that Peano Arithmetic proves Fermat's Last Theorem

So, Peano Arithmetic almost definitely proves Fermat's Last Theorem (see What is known about the relationship between Fermat's last theorem and Peano Arithmetic?). That being said, constructing an ...

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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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### 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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### 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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### 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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### 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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### 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 Integral ...

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...

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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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### 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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### 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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### 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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### 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-...

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### 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 ...

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### 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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### 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 ...

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### 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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### 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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### 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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### 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));...

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### 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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### 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) ...

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### Embedding Theorem for topological spaces, and in general

There are many examples throughout mathematics of abstracting the formal properties of a "familiar" structure, but then having a theorem stating that all models of the abstract axioms embed into one ...

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### $\epsilon$-Formalization of Undecidability of CH

Can the statement
CH is not provable in ZFC
be formalized as en $\epsilon$-Formula $\phi$ s.t. $ZFC \vdash \phi $
If so why is it refered to as an "metatheorem".

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### Applications of line graphs

I am trying to collect a few examples of applications of line graphs in sciences other than mathematics. To be more precise: I am thinking of models where there is a clear conceptual added value in ...

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### subset sum problem when the number of integers in the sum is known

Hi everyone,
I'm trying to solve a variation of the subset sum problem (http://en.wikipedia.org/wiki/Subset_sum_problem) in which all the integers that I'm using are strictly positive and (most ...

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### Godel 's Ladder: Undecidable PI_N sentences for N =2, 3, …

After Godel's groundbreaking results, a plethora of $\Pi_1^0$ undecidable arithmetical sentences have been found by many authors.
But what about $\Pi_n^0$ for $n=2,3,.....$ ?
There are, to my ...

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### Arguments against Reductio ad Absurdum [closed]

Could Reductio ad Absurdum not be consireded a valid proof method? Are there any compelling arguments against it, or at it's favor?
I feel like I am assuming some metamathematical hypothesis about my ...

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### Applicability of Deduction theorem to Primitive recursive arithmetic [closed]

Hello. I already asked the question here. The main point is that I tried to prove in Primitive recursive arithmetic (PRA) the totality of the Ackerman function, and I found, that the single thing ...

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### Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC

Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...

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### An undergraduate's guide to the foundational theorems of logic [closed]

How would you explain one of these theorems in the foundations of mathematics to a fellow colleague outside the field of logic (or rather to an undergraduate mathematics student) handwaving over the ...

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### A meta-mathematical question related to Hilbert tenth problem

I am reading Bjorn Poonen's very nice survey on Hilbert's Tenth problem
(http://www-math.mit.edu/~poonen/papers/uniform.pdf), and while I believe I understand the mathematics well, I have widespread ...