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Results tagged with set-theory
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user 3554
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
18
votes
4
answers
11k
views
How to define tuples?
As you probably know, you can define $2$-tuples $(x_1,x_2)$ as $\{\{x_1\},\{x_1,x_2\}\}$; then you can define $n$-tuples $(x_1,x_2\ldots,x_{n})$ as $((x_1,x_2\ldots,x_{n-1}),x_n)$.
In alternative, yo …
- 737
1
vote
Accepted
Is reflexivity of equality an axiom or a theorem?
I managed to write down a proof for the reflexivity of equality using only the definition of equality in terms of membership and the rules of natural deduction.
Premise: $\forall x_0\forall x_1\left …
- 737
2
votes
3
answers
778
views
Does the axiom of specification prevent writing any proof?
In set theory, the axiom of specification says that $\forall x_0\exists x_1\forall x_2\left(x_2\in x_1\leftrightarrow x_2\in x_0\land\theta\left[x_2\right]\right)$, where $\theta\left[x_2\right]$ is a …
- 737
4
votes
5
answers
3k
views
Is reflexivity of equality an axiom or a theorem?
Everybody knows that equality is reflexive: $\forall(x)(x=x)$. But should reflexivity of equality be taken as an axiom of logic or as a theorem of set theory?
If you choose the former then you probab …
- 737
4
votes
3
answers
3k
views
First-order logic without equality and set theory
Is it possible to build set theory on first-order logic without equality?
For example, how could one show that if $x_0=x_1$ then $\left\{x_0\}=\{x_1\right\}$, where $x_0$ and $x_1$ are two sets? And …
- 737