Questions tagged [foundations-of-mathematics]
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Does set-theoretic pluralism, about axiom systems, inevitably become an invitation to non-axiomatic systems of set theory?
Per Hamkins[[11][12]] (see also his [22]), if no individual axiom is too sacred to be denied in some possible world,Q and so if no collection of such axioms is so sacred either, yet then:
The ...
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Is infinity a number?
So I've been on a number of math fora, part of learning some calculus (not much of set theory, no). To my surprise I found what I would describe as strong resistance from some folks against (using) ...
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Why do we have problem of concept of set?
I am reading George Boolos's "The Iterative Concept of Set," and in the first chapter of this paper, he criticizes Cantor's definition of a set as a whole or totality of objects, pointing ...
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Need help understanding how certain mathemetical statements across the landscape can seemingly contradict (e.g. Cantor-Hume vs Euclid)
Here are the main components to my understanding on this issue:
Almost all of math can be given in a foundation of set theory
Different math can seemingly contradict, e.g. in Euclidean geometry ...
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How do skeptics explain axioms not being arbitrary?
I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
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The smallest possible formal definition of FOL
I find the common presentation of first order logic somewhat confusing. I feel that I often don’t understand why we need the exact terms and concepts we do.
My current recapitulation of “standard FOL” ...
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What does it mean to say that two theorems (provable statements) are 'equivalent'?
sometimes one sees/reads assertions such as "[the bounded inverse theorem] is equivalent to both the open mapping theorem and the closed graph theorem", but taken formally and literally this ...
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Is there a set theory which implies the interval [0, 1] but no more?
A deductive system (as a collection of judgments and rules of inference) can be used to describe something commonly called a “set theory”. We can imagine a priori there are certain properties we would ...
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Is there a limited number of 'pragmatic' logic rules?
What you have cited is a pragmatic limit, as you have not seen logic
systems with more than 8 or so precepts.
IF there were such a limit to precept quantity,
then YES there would be a limit to the ...
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Does the anticlass principle solve the Burali-Forti problem?
Justification of the foundations-of-mathematics tag: I was reading through a long text on category theory, Abstract and Concrete Categories: The Joy of Cats, and they make much of the class/set ...
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Is it a problem for arithmetic or our representation (or both) that there is incompleteness?
Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
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What is a natural number?
It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are.
I understand they are symbols at a surface level, but there is obviously more to it....
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What are the First Principles of Euclidean Geometry (Besides the Axioms)?
On first principles, Wikipedia says:
A first principle is an axiom that cannot be deduced from any other within that system. The classic example is that of Euclid's Elements; its hundreds of ...
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Do Gödel's incompleteness theorems and Tarski's theorem of indefinability of truth show we can never discover and prove every truth?
I thought I had a grasp on this. Do Gödel's apply to just math; logic, too; or more, and what does its applicability entail? If it applies to math, does it apply to physics? Similarly with Tarski: can ...
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Is mathematics analytic or synthetic?
This question is related to another question I posted but I think it requires its own treatment since of the already wide scope of the other question i.e. Is the classical theory of concepts ...
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