Questions tagged [metamathematics]
the mathematical discipline that applies mathematical methods to the study of mathematical theories themselves.
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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 $PA + \...
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The enigmatic complexity of number theory
One of the most salient aspects of the discipline of number theory is that from a very small number of definitions, entities and axioms one is led to an extraordinary wealth and diversity of theorems, ...
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Top-down mathematics, or "Where it all begins"
Sorry if this is off-topic.
It was my attempt to take a top-down approach to mathematics.
Being an inexperienced undergraduate (so please take my writing here lightly), I've been presented with ZFC as ...
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Cryptomorphisms
I am curious to collect examples of equivalent axiomatizations of mathematical structures. The two examples that I have in mind are
Topological Spaces. These can be defined in terms of open sets, ...
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Meta$^{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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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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Feferman's extensional and intensional applications of the method of arithmetization
At the very beginning of Feferman's Arithmetization of metamathematics in a general setting it can be read:
The method of arithmetization, as developed by Gödel[10], exploits the possibility of ...
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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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Can infinity shorten proofs a lot?
I've just been asked for a good example of a situation in maths where using infinity can greatly shorten an argument. The person who wants the example wants it as part of a presentation to the general ...
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Are there any good nonconstructive "existential metatheorems"?
Are there any good examples of theorems in reasonably expressive theories (like Peano arithmetic) for which it is substantially easier to prove (in a metatheory) that a proof exists than it is ...
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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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Are there examples of nonconstructive metaproofs?
This came up in a question on the xkcd forums. Is it possible to have a nonconstructive metaproof, i.e. a proof that there exists a proof in some formal system which does not construct said proof? Are ...
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"Let $x \in A$", beginning a proof of "$\forall x \in A$ ...", if A were empty [closed]
I work at a four-year teaching school, where we pride ourselves on teaching pure math, proof, and a rather obsessive carefulness of work. Recently I have been criticized for saying that "Let $x \in A$...
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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 ...
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Are there any important mathematical concepts without discrete analog?
In "continuous" mathematics there are several important notions such as covering space, fibre bundle, Morse theory, simplicial complex, differential equation, real numbers, real projective plane, etc. ...
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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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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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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 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 ...