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In Wikipedia we can find the next information

Given any set $A$, there is a set $B$ such that, given any set $x$, $x$ is a member of $B$ if and only if $x$ is a member of $A$ and $\varphi$ holds for $x$.

...there is one axiom for every such predicate $\varphi$.

https://en.wikipedia.org/wiki/Axiom_schema_of_specification#:~:text=Given%20any%20set,for%20x.

I have three questions:

(1) What is the formal and mathematical definition of a predicate?

(2) What is a predicate?

(3) What does it mean that a predicate $\varphi$ holds for a set $x$?

RataMágica
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    $(1)$ is tedious to define carefully, in terms a textual version of a statement, but it really just means a statement. It might not even depend on $x.$ You might take the sentence $\phi:1+1=2.$ Then $B={x\in A\mid 1+1=2}$ is just $A.$ The question is, then, what is a statement? It requires the whole grammar of logic, which is no fun, but you know one when you see it. – Thomas Andrews Jun 24 '25 at 05:21
  • In this context, a predicate is a formula in the first-order language of set theory, containing a free variable (call it $x$), which assigns a truth value (“true” or “false”) to any given set $x$. To use a (somewhat imperfect) programming analogy, it’s a lot like a function (or, more accurately, a C macro) that returns a boolean, such as isAwesome(x). “Predicate holds” means “predicate is ‘true’”. Books like Elements of Set Theory and A Mathematical Introduction to Logic by Enderton, or Mathematical Logic by Ebbinghaus et. al., give a good introduction to this formalism. – NikS Jun 24 '25 at 23:34
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    FWIW, I strongly disagree with the closing of this question as a duplicate of this one. The answers to the alleged duplicate are very general, not at all specific to the case of set theory that the OP is asking about. It is also (IMO) pretty obvious that the answers are given at a somewhat advanced level not appropriate for the fairly introductory background that the OP appears to be coming from. – NikS Jun 24 '25 at 23:46
  • @NikS, when you say

    "a predicate is a formula in the first-order language of set theory, containing a free variable (call it $x$), which assigns a truth value (“true” or “false”) to any given set $x$".

    First, you say $x$ is a free variable, and then you say it is a set. How is that possible?

     – RataMágica Jun 26 '25 at 00:25
  • $x$ is a variable that refers to a set. This is similar to how in programming languages you may have a variable $x$ (or whatever variable name you like) that refers to, say, an integer. Or, in a mathematical formula like $z^3 - 5z + 2$, $z$ is a variable that refers to, say, a complex number. – NikS Jun 26 '25 at 01:21

1 Answers1

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  1. A predicate is a truth function - a relation that maps arguments to values of true or false.

  2. The claim that predicate $\varphi$ holds for a set $x$, means that $x$ has the property that it evaluates $\varphi$ as true.

Graham Kemp
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  • Could you give me an exampme of a predicate? – RataMágica Jun 25 '25 at 04:19
  • ${x\in B: \lvert x\rvert=2}$ is the set of sets in $B$ for which the predicate "the set's cardinality equals $2$" holds. @RataMágica – Graham Kemp Jun 25 '25 at 23:19
  • Why is "the set's cardinality equals $2$" a truth function? It seems that "the set's cardinality equals $2$" maps objects to sentences, not to truth values – RataMágica Jun 26 '25 at 00:22
  • @GrahamKemp thank you for the edit of my answer, that's very kind of you. Actually, I've learnt the code from one of your answers and used once – Tankut Beygu Jun 27 '25 at 07:57