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Results tagged with mathematical-philosophy
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user 2000
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
29
votes
Accepted
How much should the average mathematician know about foundations?
The answer is essentially the same as how much should the average mathematician know about combinatorics? Or group theory? Or algebraic topology? Or any broad area of mathematics... It's good to know …
18
votes
Interesting meta-meta-mathematical theorems?
This answer is intended to clarify my comments to Sébastien's answer and also to propose a properly meta-meta-fact.
There is an intrinsic problem with the idea of meta-meta-theorems because theorems …
- 44.4k
12
votes
Accepted
Where are we working when we prove metamathematical theorems?
There are many flavors of "meta" in logic. Most make very minimal use of the metatheory. For example, the Montague Reflection Principle in Set Theory says the following:
Metatheorem. For every fo …
- 44.4k
3
votes
Accepted
In what sense Fraissean view point shows Model Theory can be done without any formal syntax ...
After your answer, I think I understand better where you see a problem. I don't think you fully appreciate the way of interpreting formulas from Fraïssé's point of view. For simplicity, I will follow …
- 44.4k
9
votes
Unprovable sentence about integers
As noted in the comments, the Paris–Harrington Theorem is a natural example of a true statement that is not provable in Peano Arithmetic. In fact, the Paris–Harrington Theorem is equivalent over PA to …
- 44.4k
7
votes
Are there natural examples of mathematical statements which follow from consistency statements?
The Paris–Harrington Theorem is equivalent (over IΣ1) to Con(PA + Tr(Π1)), where Tr(Π1) is the set of all true Π1 sentences in the language of arithmetic.
For clarity, the 1-consistency of PA, i.e. w …
- 44.4k
12
votes
Accepted
Intended interpretations of set theories
While Kunen takes for universe the collection of all hereditary sets, Marc proposes to restrict the universe to those hereditary sets which are first-order definable without parameters (Marc's "provab …
- 44.4k
7
votes
Can you prove equivalence without being able to calculate it?
Computability Theory has many examples of this. For example, the halting set $K$ is order-isomorphic to $\mathbb{N}$, but there is no computable order-preserving bijection between them.
39
votes
Why is this new result such a big deal?
The statement in question, frequently denoted $\mathsf{RT}^2_2$ in the context of reverse mathematics, is the instance of the infinite Ramsey theorem for unordered pairs and two colors. Specifically, …
- 44.4k
28
votes
Nelson's program to show inconsistency of ZF
Edward Nelson passed away at age 82 on September 10, 2014. You can read a tribute to Nelson's illustrious career from Princeton University.
Although he worked on the inconsistency of PA until the end …