Talk:Axiom of infinity

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Mathematics rating: Start Class Low Priority  Field: Foundations, logic, and set theory

Q: aren't capital names and variables reserved for second-order logic? \exists x \lbrack \emptyset \in x \land \forall y(y \in x \rightarrow \cup \lbrace y, \lbrace y \rbrace \rbrace \in x) \rbrack --Alterego 01:54, Apr 25, 2005 (UTC)

If you want to quantify both over first-order and second-order things you might want to make that convention or another which makes the distonction clear. Since we are working entirely in first-order here we don't need to restrict ourselves as to the variables which we use. Using a variable name which has the right associations can help people a lot in understanding a formula. MarSch 12:13, 27 Apr 2005 (UTC)
I don't want to, but the article does. I guess it is a convention of set builder notation? --Alterego 15:11, Apr 27, 2005 (UTC)

[edit] Specification to remove unwanted elements?

What specification are you using to remove unwanted elements?

I am not sure exactly what you mean by using the axiom schema of specification to remove unwanted elements. The schema allows one to remove elements from a set which do not satisfy a particular predicate. However as far as I can see, there is no such predicate applicable in this case.

In fact, the "standard" way to do this is to just take all inductive subsets of some inductive set and find their intersection, but even this I believe to be fundamentally flawed (I'll not go into this here, though). Kidburla2002 9 July 2005 12:45 (UTC)

I'm not sure I fully understand what you're getting at. That said, it's clear that you're basing your discussion on intuitive set theory, which is not what this article is about. This article is about axiomatic set theory, which starts with first principles defined abstractly, regardless of pre-theoretic intuitions. The most common version is Zermelo-Fraenkel set theory (ZF), which is a simple axiomatic theory, expressed in predicate logic, of what the symbol \in means. ZF is a single-sorted theory, i.e., it makes no distinction between set-variables and objects and non-set variables and objects. ZF simply says that certain sets exist (e.g. the empty set) and how to form sets from existing sets (e.g. pairing). It turns out that for finite, well-founded sets, ZF matches everyone's intuitions about sets quite nicely. But it's precisely the various kinds of infinite sets that it's hard to have pre-theoretic intuitions about: everybody has encountered finite collections of physical objects, but nobody has encountered infinite collections, so it's a bit hard to have intuitions about how they should behave. The axiomatic approach is useful because it makes precise how all sets must behave, without having to rely on intuitions which people can disagree about. Most of the axioms say trivial things (e.g., "there is an empty set", which, by the way, does not mean the same as "there is no set" as you suggest) which are necessary for an approach based on logic where you want to be able to prove everything from first principles. The present axiom of infinity simply says that there exists a set with a certain recursive property. (It doesn't actually say that the set whose existence is postulated is infinite or is related in any way to the natural numbers. These notions require additional definitions and conventions.) The recursive property is explained in the article: the infinite set is such that if it contains a set x, then it also contains what you might call the successor of x (which is the set x\cup\{x\}). Note that the existence of sets which contain the empty set and finitely many successors follows from the other axioms of ZF. But it is not possible to prove the existence of a set which contains the empty set and all of its successors from the remaining axioms of ZF, hence the need for the present axiom which explicitly asserts that such a set exists. --MarkSweep (call me collect) 08:04, 29 April 2006 (UTC)
You might be missing the point here. The article says
This set S may contain more than just the natural numbers, forming a subset of it, but we may apply the axiom schema of specification to remove unwanted elements, leaving the set N of all natural numbers.
To paraphrase: "Infinity asserts the existence of a superset of omega, separation can be used to get exactly omega." What predicate can be used to do this separation? -Dan 17:18, 24 May 2006 (UTC)

ω is the set of natural numbers. n is a natural number iff \lbrack n = 0 \or \exists y(n = y\prime) \rbrack \and \forall x \in n \lbrack x = 0 \or \exists y \in n(x = y\prime) \rbrack \!. That is, n is a natural number iff it is either zero or a successor and each of its elements is either zero or a successor of another of its elements. This uses the axiom of regularity. JRSpriggs 06:34, 26 June 2006 (UTC)

There is more about this at Talk:Natural number#Set theoretic definition. JRSpriggs 04:55, 1 July 2006 (UTC)

[edit] what does this symbol mean?

What does \forall k \in n(\bot) mean? What is the \bot symbol? -lethe talk + 16:55, 8 March 2007 (UTC)

\bot is a logical symbol that means "false". It is often taken as an atomic formula that is identically false. So \forall k \in n(\bot) means n = \varnothing. In a theory with equality, \bot can be simulated by not equals: n = \varnothing \Leftrightarrow \forall k \in n(\bot) \Leftrightarrow \forall k \in n ( k \not = k).
This article, like many of the "axiom of ..." articles, could use a good cleaning. CMummert · talk 17:25, 8 March 2007 (UTC)