What's the meaning of an axiom in SGA?

Asked Dec 16 '21 at 10:55

Active Dec 16 '21 at 11:01

Viewed 58 times

In SGA 4-1, Grothendieck stated the following axiom:

($\mathscr{U}$ B) Soit $R\{x\}$ une relation et $\mathscr{U}$ un univers. S'il existe un élément $y\in\mathscr{U}$ tel que $R\{y\}$, alors $\tau_{x}R\{x\} \in\mathscr{U}$.

However, I don't know the meaning of the symbols $R\{x\}$ and $\tau_{x}R\{x\}$. I guess that $R\{x\}$ stands for a subset of $x\times x$, since it says that $R\{x\}$ is a relation.

edited Dec 16 '21 at 11:01

asked Dec 16 '21 at 10:55

Toney Leung

2

See this MO-post for references. – Dietrich Burde Dec 16 '21 at 11:07
1

see also this post for the symbol – Mauro ALLEGRANZA Dec 16 '21 at 11:14
1

In a nuthsell the first one means a relation $R(x)$ (e.g. "x is Rational") while the second one corresponds to existential quantification: $\exists x R(x)$. – Mauro ALLEGRANZA Dec 16 '21 at 11:23
1

The second is a term, a selector, designating some $x$ such that $R(x)$, when one exists. Bourbaki uses $\tau$ for this; Hilbert used $\sigma$. – BrianO Dec 16 '21 at 20:04

What's the meaning of an axiom in SGA?

0 Answers0