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Results tagged with mathematical-philosophy
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user 6794
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.
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History of Logic Development
Although it's not exactly what you asked for, you might take a look at the book "Foundations of Mathematics" by William S. Hatcher. It's primarily about the various foundational systems themselves, b …
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Accepted
What is against having distinct membership relations on sets in the Platonic realm?
Putting on my Platonist hat for a while (and it's a very comfortable hat), I'd answer the question as follows: There's nothing wrong with having and studying two or more relations, like $\in_1$ and $\ …
- 73.2k
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Where do models of false theories exist?
I'll answer this from a Platonic viewpoint. Consider a theory $T$ that is false when its primitive concepts are interpreted in the standard way. For example, $T$ could be the theory ZF$\neg$C that you …
- 73.2k
8
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The universe of sets, existential quantification in set theory
I regard ZF (or better ZFC) as a (partial) description of the behavior of actual sets. The theorem you quoted says, in that context, that there is no set containing everything. In the same context, …
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Is it natural to hold that Ur-elements, small & big sets and proper classes exists?
Well-foundedness is part of what I mean by "set". (Extensionality is another part.) And ZFC summarizes some aspects of my intuitive conception of sets.
Certainly other sorts of things exist. For exa …
- 73.2k
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Proper classes and their consequences
A fairly general "definition" of "proper class" is that it means a collection of sets that is not itself a set.
In the usual picture of sets as constituting a transfinite cumulative hierarchy (in w …
- 73.2k
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Accepted
Can finite sets be non-c.e. depending on how they are presented?
Burgin seems to conflate the notion of computability (existence of an algorithm) with a stronger notion such as our knowing an algorithm or existence of a proof that a particular algorithm agrees with …
- 73.2k
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Why is it OK to rely on the Fundamental Theorem of Arithmetic when using Gödel numbering?
Stefan has explained the essential points well, but let me add some details and a reference. For the first incompleteness theorem, one can work with a very weak theory of arithmetic. One needs to be …
- 73.2k
18
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How to tell a paradox from a "paradox"?
Both the Russell paradox and the Banach-Tarski "paradox" show that certain ideas are contradictory. It seems to me that the key difference between the two is that, in Russell's case, the ideas in que …
- 73.2k
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Are proper classes objects?
Proper classes are not objects. They do not exist. Talking about them is a convenient abbreviation for certain statements about sets. (For example, $V=L$ abbreviates "all sets are constructible.") …
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34
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Interpretation of the Second Incompleteness Theorem
For the philosophical point encapsulated in (*) in the question, it seems that corollaries of the second incompleteness theorem are more relevant than the theorem itself. If we had doubts about the c …
- 73.2k
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nonstandard models and mathematical theorems
Well, let's compare the compactness example you cited with a non-standard models approach to the same result. Of course, since the result is about vertices of a graph, not just natural numbers, I'll …
- 73.2k
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Tarski-Grothendieck in the cumulative hierarchy
My intuition supporting a belief in a proper class of (strongly) inaccessible cardinals (equivalently, Grothendieck universes) is that, since the cumulative hierarchy is intended to continue "forever" …
- 73.2k
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Why is the notion of algorithm a primitive one in Brouwer's intuitionism?
As far as I know, Brouwer's intuitionism involves a primitive notion of "construction". This might be viewed as "a finite routine" --- finite because it should be possible to finish a construction. I …
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What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to cap...
The axioms that you listed are satisfied in the cumulative hierarchy over any set of atoms. That is, begin with some entities that are not sets, for example people. Then consider (1) all sets of those …
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