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Results tagged with mathematical-philosophy
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user 1946
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.
19
votes
Accepted
What governs our "perception?" about the platonic realm of sets?
The recent developments on the consistency of NF bring welcome closure to the longstanding open question about whether NF was consistent. And this is naturally a very important matter for those who fi …
- 236k
8
votes
Accepted
Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.
In a bottomless model of ZFC, the mantle is not a ground. It is the …
- 236k
11
votes
Silver's approach to the inconsistency of $\mathrm{ZFC}$
When I was a graduate student at Berkeley in the early 90s, I had heard that Silver's approach to refuting ZFC involved the idea that somehow we make a mistake in our thinking about ZFC by conflating …
- 236k
8
votes
Mathematical analysis of Lewisian concepts, esp. natural properties
I have engaged with the Lewis-style set-theoretic mereology in a few papers, undertaken jointly with Makoto Kikuchi. My interest in this topic was inspired originally by a MathOverflow question, Why h …
- 236k
23
votes
Accepted
Are the categories of sets, abelian groups, and commutative rings unique?
Introduction to pluralism
A version of this question lies at the heart of the ongoing dispute on pluralism in the philosophy of mathematics. Is there at bottom just one mathematical reality? Does ever …
2
votes
Quantification over uncountable sets
There are several things one can say.
The theory of ZFC without powerset is often denoted by $\newcommand\ZFCm{\text{ZFC}^-}\ZFCm$. One has to be a little careful with what it means, since collection …
- 236k
21
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I believe that there are many instances of this phenomenon in set theory, where an elaborate theory is developed over a period of years by many people, even though the theory is not viewed ultimately …
26
votes
Siegel zeros and other "illusory worlds": building theories around hypotheses believed to be...
I have heard that Jack Silver's discovery of zero sharp ($0^\#$) was part of his attempt to show measurable cardinals inconsistent. Instead of finding the long-sought-after contradiction, however, he …
5
votes
Accepted
Are ITTM's necessary to compute Turing's "computable numbers" and what does that mean for or...
No, classical computability theory as you point is quite capable of dealing with infinitary computable enumerations and computability-in-the-limit from its earliest stages. I believe that Turing is to …
- 236k
18
votes
Why not adopt the constructibility axiom $V=L$?
Although the axiom of constructibility is often resisted by set theorists with the view that it is restrictive, nevertheless there are a variety of ways in which the axiom is compatible with strength …
20
votes
Axiom of Choice versus V=L in opposition to large cardinals
Consider the relativized constructibility hypothesis, which asserts that $V=L[A]$ for some set $A$.
This axiom is compatible with any locally verifiable large cardinal property, properties that can …
- 236k
15
votes
1
answer
965
views
Does every model of ZF-foundation have an extension, with no new well-founded sets, where ev...
This question follows up on an issue arising in Peter LeFanu
Lumsdaine's nice question: Does foundation/regularity have any
categorical/structural consequences, in
ZF?
Let me mention first that my vi …
- 236k
48
votes
Contemporary philosophy of mathematics
Let me mention a few current issues on which I have been involved in the philosophy of
mathematics. Of course there are also many other issues on which people are working.
Debate on pluralism. First, …
5
votes
What is against having distinct membership relations on sets in the Platonic realm?
It turns out that one cannot have two fundamentally different parallel set membership relations $\in$ and $\in^*$, if they both satisfy ZF with respect to the common language, for in this case they mu …
- 236k
7
votes
Accepted
Is the statement "All numbers are counting numbers" independent of $PA$?
The statement asserting that every number is a counting number is $\forall n\ C(n)$, and this is definitely independent of PA, if PA is understood to include induction only in the usual language of ar …
- 236k