3
votes
2answers
425 views

Are descriptive and ontological notions of equality equal? [closed]

‎Let ‎$‎‎a$ ‎and ‎‎$‎‎b$ ‎are ‎two "‎objects". ‎What ‎is ‎the ‎meaning ‎of‎ ‎‎$a=b‎‎$‎? This is one of the deepest problems of philosophy and logic because one needs a complete information about ...
10
votes
5answers
1k views

Intended interpretations of set theories

In his Set Theory. An Introduction to Indepencence Proofs, Kunen develops $ZFC$ from a platonistic point of view because he believes that this is pedagogically easier. When he talks about the intended ...
11
votes
2answers
533 views

Inconsistency and workaday independence.

Set-theoretic topologists, for example, encounter many propositions that turn out independent from set theory. Sometimes these results require novel forcing arguments, but often they simply rely on ...
19
votes
5answers
2k views

Supervenience in mathematics

I'm not quite sure if this is the right place to ask, and if this is the right way to ask, but I dare. In philosophy (of mind, e.g.) the concept of supervenience is used: "Supervenience [is] used ...
15
votes
2answers
2k views

Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides: http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf particularly, to the "Universe view simulated inside Multiverse", p. 22. My question is: is it very unsound ...
16
votes
7answers
4k views

What is Realistic Mathematics?

This post is partially about opinions and partially about more precise mathematical questions. Most of this post is not as formal as a precise mathematical question. However, I hope that most readers ...
9
votes
2answers
1k views

Proving Independence of Axioms by Exhibiting Models Which Don't Satisfy Our Intuition

I recently saw the proof of the independence of ZF (with allowance for multiple empty sets) and AC. The proof constructed the model based on a set theory generated by infinitely many empty sets and ...
5
votes
5answers
3k views

Models of ZFC Set Theory - Getting Started

For just any first-order theory: What are the sets I am supposed/allowed to think of when thinking of models as sets (of something + additional structure)? Provided: I can think of models of any ...
0
votes
2answers
398 views

In what sense Fraissean view point shows Model Theory can be done without any formal syntax and deduction rule?

In this post I want to look at an issue I was in doubt when looking at the comment of F. G. Dorais in the post In model theory, does compactness easily imply completeness? F. G. Dorais remark was: ...