Set Theory¶
Job: Axiomatize mathematics in ZFC; track ordinals/cardinals; record independence phenomena (AC, CH) as first-class research facts.
Core¶
Language: \(\in\) (and \(=\)). ZFC axioms (schematic where needed):
| Axiom | Role |
|---|---|
| Extensionality | sets determined by elements |
| Empty set / Pairing / Union | basic constructors |
| Power set | \(\mathcal{P}(x)\) exists |
| Infinity | inductive set \(\Rightarrow\omega\) |
| Separation | \(\{y\in x:\varphi(y)\}\) |
| Replacement | image of a set under a class function is a set |
| Foundation | \(\in\)-well-founded; no infinite \(\in\)-descent |
| Choice (AC) | every family of nonempty sets has a choice function |
Ordinals: transitive sets well-ordered by \(\in\). \(\omega\), successor \(\alpha\cup\{\alpha\}\), limits. Transfinite induction/recursion along ordinals.
Cardinals: initial ordinals; \(|A|\le|B|\) iff injection \(A\to B\). Cantor: \(|A|<|\mathcal{P}(A)|\). \(\aleph_0,\aleph_1,\ldots\); continuum \(2^{\aleph_0}=|\mathbb{R}|\).
Well-ordering theorem \(\Leftrightarrow\) AC \(\Leftrightarrow\) Zorn \(\Leftrightarrow\) every vector space has a basis (over ZF; some equivalences need care).
Independence awareness. - Gödel (1938): \(\mathrm{Con}(\mathrm{ZF})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\mathrm{GCH})\) via constructible universe \(L\); CH holds in \(L\). - Cohen (1963): forcing \(\Rightarrow\) \(\mathrm{Con}(\mathrm{ZF})\Rightarrow\mathrm{Con}(\mathrm{ZFC}+\neg\mathrm{CH})\) (and many other configurations of \(2^{\aleph_0}\)). - Thus CH is independent of ZFC (relative consistency). AC is independent of ZF (Fraenkel–Mostowski / Cohen). - Large cardinals, determinacy, forcing axioms (MA, PFA) decide many natural statements beyond ZFC but are additional commitments.
Reflection / hierarchy: \(V_\alpha\) cumulative hierarchy; \(V=\bigcup_\alpha V_\alpha\) under Foundation. Most mathematics lives in \(V_{\omega+n}\) or with modest ordinals; set theory studies \(V\) globally.
Universes for category theory (pointer): Grothendieck universes / NBG / MK class-set theories avoid size paradoxes for \(\mathbf{Set}\).
See it¶
Forcing idea (one line): add a generic filter \(G\) through a poset \(P\in M\) to form \(M[G]\) with controlled new subsets of \(\omega\) (e.g. add \(\aleph_2\) many reals without collapsing \(\aleph_1\) carefully) so \(M[G]\models 2^{\aleph_0}>\aleph_1\).
Check¶
Does 'undecidable in ZFC' mean false?
No. Independence means ZFC proves neither \(\varphi\) nor \(\neg\varphi\) (if Con(ZFC)). Mathematicians may adopt extra axioms or work relatively.
Why Replacement is not implied by Separation alone.
Separation only carves subsets of existing sets. Replacement produces sets the size of images of large domains (e.g. \(\{V_{\omega+n}:n\in\omega\}\) issues / Hartogs-type constructions). Without Replacement, ordinal arithmetic and cardinal comparison are crippled.
Pitfalls¶
- Naive comprehension \(\{x:\varphi(x)\}\) \(\Rightarrow\) Russell paradox; Separation restricts to subsets.
- Continuum function \(\kappa\mapsto 2^\kappa\) is almost free under Easton forcing for regulars (with constraints); not arbitrary for singulars (Shelah).
- Foundations debates (type theory, HoTT, NF) exist; ZFC remains the default mathematical base.
Refs¶
Kunen, Set Theory; Jech; Kunen/Jech forcing chapters; Kanamori, The Higher Infinite; Halbeisen.