Countability: Difference between revisions

Line 3:

<nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki>

Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the continuum hypothesis. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven.

Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the [[continuum hypothesis]]. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven.

Revision as of 17:38, 3 September 2023 (edit) RhubarbJayde (talk \| contribs) No edit summary Tag: Visual edit ← Older edit		Revision as of 17:52, 3 September 2023 (edit) (undo) RhubarbJayde (talk \| contribs) No edit summary Tag: Visual edit Newer edit →
Line 3:		Line 3:
	<nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers\|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki>		<nowiki>In terms of ordinals, it is clear that \( \omega \) is countable. You can also see that \( \omega + 1 \) is countable, by pairing \( \omega \) with zero and \( n \) with \( n + 1 \); that \( \omega 2 \) is countable, by pairing \( n \) with \( 2n \) and pairing \( \omega + n \) with \( 2n + 1 \), and so on. Larger countable ordinals such as \( \omega_1^{\mathrm{CK}} \) also are countable but, due to their size, such a map \( f \) is not \( \Delta_1 \)-definable. Furthermore, a gap ordinal may have a map to \( \mathbb{N} \) but this map can not be defined at all using first-order set theory. The least uncountable infinite ordinal is denoted \( \omega_1 \) or \( \Omega \), and it is larger than anything that can be reached from \( \omega \) using successors and countable unions. In particular, it is an </nowiki>[[Epsilon numbers\|epsilon number]]<nowiki>, and much more. This is why it is useful as a "diagonalizer" in the construction of ordinal collapsing functions, although \( \omega_1^{\mathrm{CK}} \) is sometimes used instead.</nowiki>

	Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the continuum hypothesis. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven.		Since \( \omega_1 \) and \( \mathbb{R} \) are both uncountable, it is natural to ask whether they have the same size. The affirmative of this question is known as the [[continuum hypothesis]]. Cantor failed to prove or disprove it, and Gödel and Cohen later proved that, if the \( \mathrm{ZFC} \) axioms are consistent, then the continuum hypothesis can neither be proven nor disproven.

Countability: Difference between revisions

Navigation menu

Search