Content deleted Content added
→Mahlo cardinals and reflection principles: wouldn't it be weird if the fixed point theorem could imply a Mahlo cardinal but ZFC can't? i'm strongly inaccessibly far from being an expert but I'm failing to fact check these statements |
Adding example of author using this convention →α-Mahlo, hyper-Mahlo and greatly Mahlo cardinals |
||
| (4 intermediate revisions by 3 users not shown) | |||
Line 1:
{{Short description|Concept in mathematics}}
In [[mathematics]], a '''Mahlo cardinal''' is a certain kind of [[large cardinal]] number. Mahlo cardinals were first described by {{harvs|txt|authorlink= Paul Mahlo|first=Paul|last=Mahlo|year1=1911|year2=1912|year3=1913}}. As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by [[ZFC]] (assuming ZFC is [[consistent theory|consistent]]).
Line 40 ⟶ 41:
The term α-Mahlo is ambiguous and different authors give inequivalent definitions. One definition is that
a cardinal κ is called α-Mahlo for some ordinal α if κ is strongly inaccessible and for every ordinal β<α, the set of β-Mahlo cardinals below κ is stationary in κ.<ref>W. Boos, ''[https://www.proquest.com/openview/22276484b3832b4d487425ee882b5b09/1 Nonstandard Large Cardinals]'' (1971).</ref><sup>p. 3</sup> However the condition "κ is strongly inaccessible" is sometimes replaced by other conditions, such as "κ is regular" or "κ is weakly inaccessible" or "κ is Mahlo". We can define "hyper-Mahlo", "α-hyper-Mahlo", "hyper-hyper-Mahlo", "weakly α-Mahlo", "weakly hyper-Mahlo", "weakly α-hyper-Mahlo", and so on, by analogy with the definitions for inaccessibles, so for example a cardinal κ is called hyper-Mahlo if it is κ-Mahlo.
A regular uncountable cardinal κ is '''greatly
For α < κ<sup>+</sup>, define the subsets M<sub>α</sub>(κ) ⊆ κ inductively as follows:
*M<sub>0</sub>(κ) is the set of regular cardinals below κ,
*M<sub>α+1</sub>(κ) is the set of regular λ < κ such that M<sub>α</sub>(κ) ∩ λ is stationary in λ,
*for limits α with cf(α) < κ, M<sub>α</sub>(κ) is the intersection of M<sub>β</sub>(κ) over all β < α, and
*for limits α with cf(α) = κ, pick an enumeration f : κ → α of a cofinal subset. Then, M<sub>α</sub>(κ) is the set of all λ < κ such that λ ∈ M<sub>f(γ)</sub>(κ) for all γ < λ.
Although the exact definition depends on a choice of cofinal subset for each α < κ<sup>+</sup> of cofinality κ, any choice will give the same sequence of subsets modulo the nonstationary ideal.
For δ ≤ κ<sup>+</sup>, κ is then called '''δ-Mahlo''' if and only if M<sub>α</sub>(κ) is stationary in κ for all α < δ. A cardinal κ is κ<sup>+</sup>-Mahlo if and only if it is greatly Mahlo.
The properties of being inaccessible, Mahlo, weakly Mahlo, α-Mahlo, greatly Mahlo, etc. are preserved if we replace the universe by an [[inner model]].
Line 68 ⟶ 79:
Axiom F is the statement that every normal function on the ordinals has a regular fixed point. (This is not a first-order axiom as it quantifies over all normal functions, so it can be considered either as a second-order axiom or as an axiom scheme.)
A cardinal is called Mahlo if every normal function on it has a regular fixed point{{citation needed|date=December 2022}}, so axiom F is in some sense saying that the class of all ordinals is Mahlo.{{citation needed|date=July 2023}} A cardinal κ is Mahlo if and only if a second-order form of axiom F holds in ''V''<sub>κ</sub>.{{citation needed|date=July 2023}} Axiom F is in turn equivalent to the statement that for any formula φ with parameters there are arbitrarily large inaccessible ordinals α such that ''V''<sub>α</sub> reflects φ (in other words φ holds in ''V''<sub>α</sub> if and only if it holds in the whole universe) {{harv|Drake|1974|loc=chapter 4}}.
==Appearance in Borel diagonalization==
| |||