elementary set theory - What is the definition for an infinite set ...
2020年12月3日 · An infinite set is a set that is not a finite set. So you could show that there does not exist a bijection between your "infinite" set and a finite set. Other ways: A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number.
elementary set theory - What do finite, infinite, countable, not ...
Clearly every finite set is countable, but also some infinite sets are countable. Note that some places define countable as infinite and the above definition. In such cases we say that finite sets are "at most countable".
set theory - Why is the infinite set from the axiom of infinity the ...
2016年3月21日 · Now you also need to observe that the infinite set guaranteed by the axiom of infinity is not $\Bbb N$. It is an inductive set, which may or may not be $\omega$ (the least infinite ordinal). $\omega$ can be used to model the natural numbers internally to set theory. But we can also choose different interpretations.
Cardinality of cartesian product of an infinite set with N
Here the problem 1.4.26 from Sohrab, Basic Real Analysis Duplicate This are not duplicate of this question: Cardinality of Cartesian Product of Uncountable Set with Countable Set the answer give...
set theory - The existence of the power set of an infinite set ...
2020年8月15日 · A large part of the set theory is devoted to infinities of various kinds, and this has been built on Cantor's groundbreaking work on uncountable sets. However, even Cantor's proof is based on the assumption that certain sets exist, namely that the power set of …
An exercise about infinite set in《The Foundations of Analysis：A ...
2024年11月6日 · Other books use “a set is infinite if and only if it is equipotent to a proper subset” to determine if a set is infinite.This exercise mentioned the conclusion about the union of finite sets . Other than that, there is no conclusion about the …
Second power of an infinite set is Dedekind-infinite
2023年10月23日 · 'Infinite set' could mean a Tarski-infinite set or a set not equinumerous to any natural number. A set S is Tarski-finite iff every non-empty family of subsets of S has a $\subseteq$-maximal element. n is a natural number iff n is the empty set or n is a successor ordinal whose non-empty members are successor ordinals.
Does maximum or supremum of an infinite set exit?
An infinite set which is bounded above always has a supremum by the completeness of $\mathbb{R}$; i.e. any set which has an upper bound has a least upper bound.
Linear Independence of an infinite set - Mathematics Stack Exchange
2013年2月24日 · $\begingroup$ I cant understand @Paul why linear independence follows from the fact that "A forms an orthogonal set with respect to the inner product". Do you mind elaborating the reason. Do you mind elaborating the reason.
Maximum Number in an Infinite set - Mathematics Stack Exchange
2024年8月12日 · $\begingroup$ Note that "infinite set of random integers" is not a well-defined statement -- you need to state where these integers come from to give it meaning. $\endgroup$ – usul Commented Jun 16, 2013 at 23:59