All Questions
Tagged with philosophy elementary-set-theory
48
questions
1
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129
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Why do we need a metatheory if we can include self-referencing language in the object theory?
I am wondering why we need to have a metatheory in order to talk about a theory- why can't we just add self-referencing terms to the language of the formal system on which the theory itself is based, ...
- 3,019
2
votes
2
answers
161
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Is everything an object in Math, just like in Objected-Oriented Programming? (Tao's Analysis I)
I am reading Tao's Analysis I, and there are a number of passages which seem to suggest an object-oriented point of view of mathematics reminiscent of the object-oriented programming with which I, as ...
- 3,019
2
votes
4
answers
412
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How to interpret what a set is to see how it could be infinite?
Currently, 'infinite set' sounds oxymoronic to me, so my question is how to interpret what a set is such that it is consonant with it being infinite. I understand that we take it as axiomatic that ...
- 3,019
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0
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61
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in modal set theory, why it is issue?
I have been studying the Iterative Set Concept within the context of the paper titled "Modal Set Theory" from Menzel specifically on pages 11-12.
"As we’ve just seen, the iterative ...
- 33
3
votes
4
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764
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what is the exact meaning of the identity relation? [closed]
I would like some clarification regarding the exact meaning of the identity relation. Specifically, if $a=b$, does this mean
$(1)$ $a$ is the very same object as $b$
or does it leave open the ...
- 6,099
2
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1
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370
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Why are we confident in the ability of ZFC to formalise mathematics if very few proofs are actually converted into ZFC?
$\mathsf{ZFC}$ is often introduced in logic textbooks as a first-order theory with equality and a single non-logical symbol $\in$. However, even stating the axioms of $\mathsf{ZFC}$ in this language ...
- 20.7k
4
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2
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230
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Proof that axiom of replacement won't generate a set going arbitrarily high in the cumulative hierarchy, without relying on replacement?
Is it possible to prove, in first-order ZFC, that the axiom of replacement can't generate a set that includes members going arbitrarily high in the cumulative hierarchy, but without using the axiom of ...
- 117
0
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2
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356
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How do we understand a 'universe' in the context of Mathematics?
I recently open a can of worms for myself by inquiring if there is a difference between a number as a natural or real, and got a fair answer, in doing so I came by an interesting idea about viewing ...
- 745
11
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4
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573
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Why is it not important what mathematical objects are?
In axiomatic set theory the term "set" and the relation "$\in$" are primitive notions. Thus, it is not defined what sets are nor what the relation is. Axiomatic set theory is ...
- 389
4
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2
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229
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Are propositions sets of possible worlds?
In article "Against Set Theory" by Peter Simons (Appeared in Johannes Marek and Maria Reicher, eds., Experience and Analysis. Proceedings of the 2004 Wittgenstein Symposium. Vienna: öbv&...
- 4,631
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125
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Universal quantifier over an uncountable set
To prove that a segment has the same number of points with half a segment one might say that one can find a bijective function mapping every point from the segment to the half segment. Let' say:
$$\...
- 19
0
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149
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Formalism and interpretation
As far as I know formalism is the view that mathematics is just a manipulation of strings according to established manipulation rules, which can be thought of as a game having its rules. In set theory ...
- 439
1
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1
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186
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Interpretations of sets and set membership
I apologize in advance if anything I say in the following is incorrect and I appreciate any corrections if anything is incorrect. My knowledge of set theory and logic is extremely limited, however, I ...
- 439
0
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1
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442
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Is it possible for a set to contain non-mathematical objects?
This might be a somewhat philosophical question and perhaps I even have a wrong understanding of what I write as a premise, so I am sorry if that is the case. A set is usually any collection of ...
- 389
4
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1
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114
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Defining a set connotatively or denotatively.
“It is said that a set can be defined connotatively or denotatively. Which of these terms applies to the definition by roster and which to the definition by a defining sentence?”- p. 137, Elements of ...
- 59