# Tagged Questions

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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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### What is an example of a non-axiomatic mathematical system? [closed]

In this wikipedia article on the foundations of mathematics, it says: In practice, most mathematicians ... do not work from axiomatic systems Is this correct? If so, what is an example of this?
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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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### In What Sense is Set Theory a 'Foundation' for Mathematics? [closed]

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "theory" in scare ...
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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 ...
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### Complete mathematics

Hello, I would like to ask you if there is a mathematical theory, that is complete (in the sense of Goedel's theorem) but practically applicable. I know about Robinson arithmetic that is very limited ...
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### Bourbaki theory of isomorphism, examples of untransportable formulas

In their book "Theory of sets" Bourbaki suggested a general theory of isomorphism. (See also http://www.tau.ac.il/~corry/publications/articles/pdf/bourbaki-structures.pdf ) The example of an ...
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### Intuitive and/or philosophical explanation for set theory paradoxes

Every student of set theory knows that the early axiomatization of the theory had to deal with spectacular paradoxes such as Russel's, Burali-Forti's etc. This is why the (self-contradictory) ...
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### The isomorphism inference rule

Suppose we are writing very detailed proofs, absolutely without any gaps (for example, for checking proofs by computer). In such formal proofs every step (even a trivial one) must be justified. For ...
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### Formal definition of 'useful' ?

Has anyone worked out a formal, general-enough definition of what is 'useful', so that it could reflectively be used in mathematics? I am aware of the work in utility theory from economics (but ...
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### Is there a formal notion of what we do when we 'Let X be …'?

This is likely an elementary question to logicians or theoretical computer scientists, but I'm less than adequately informed on either topic and don't know where to find the answer. Please excuse the ...
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### Is there a known way to formalise notion that certain theorems are essential ones?

Suppose You ask a question beginning from "Why some structure is..." or "Why some object has property..." and several answers arises. Which criteria do You use to qualify which answer is correct? ...
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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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### Deficiency of necessary conditions

Motivation Consider the situation: You know that every $x$ that has property $P$ must have property $Q$. $Q$ is a rather strong condition but not strong enough to fulfill $P$. What is ...
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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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### Given is “model”. How many theories may it be a model?

Usually we have axiomatic theory and the we look for model for it - this is book picture. Of course in real math usual one has a "model" that is given structure and looks for proper axiomatizing of ...
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### Bourbaki's epsilon-calculus notation

Bourbaki used a very very strange notation for the epsilon-calculus consisting of $\tau$s and $\blacksquare$. In fact, that box should not be filled in, but for some reason, I can't produce a \Box. ...
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### Classification of properties of structures

Is there a sensible classification of the properties of structures with a given signature $\sigma$, e.g. graphs with $\sigma = \lbrace R \rbrace$? For example like this: properties defined by first-...