/prog/ is an exciting place

Every day, every hour, every minute our world is changing. Each and every one of us takes part in this change. We observe it, we react to it, we cause it. We all come from different parts of this world and we all experiencing a different facet of existence. We all have our own story, our own path from which we came, and yet we all made our way here, to /prog/. No matter where we come from, we all participate in this message board, this evolving world wide programming community. This is why /prog/ is great.

/prog/ will be spammed continuously until further notice. we apologize for any inconvenience this may cause.

The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment. The work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is commonly used as a foundational system, although in some areas category theory is thought to be a preferred foundation.

Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either κ or μ is infinite and both are non-zero, then




   
    κ · μ = max(κ, μ)

Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size. The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite.

From this definition, it is clear that a set is a subset of itself; for cases where one wishes to rule out this, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but B is not a subset of A.

Set-theoretic topology studies questions of general topology that are set-theoretic in nature or that require advanced methods of set theory for their solution. Many of these theorems are independent of ZFC, requiring stronger axioms for their proof.

There is a set A such that for all functions f (on the set of non-empty subsets of A), there is a B such that f(B) does not lie in B.

König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals. (The reason for the term "colloquially", is that the sum or product of a "sequence" of cardinals cannot be defined without some aspect of the axiom of choice.)