0

I am aware that De Margan's Morgan's laws are only "three-quarters" true in intuitionistic logic, that is, $\neg(p\wedge q)\not\vdash\neg p\vee\neg q$ (here's a simple topological model in which this fails).

This suggests to me that $(A\cap B)^c = A^c\cup B^c$ might also fail in some "intuitionistic" model of set theory, for the usual proof uses LEM. Does someone know of such a model?

Even if it holds in all the models, I understand that it might be unprovable. In this case, do constructive mathematicians just discard this law? I guess this makes their life super hard for for almost every other result, I invoke this innocuous statement.

By "set theory", feel free to assume ZF or ZFC.

Hanul Jeon
  • 28,245
Atom
  • 4,620
  • 1
    Pick your favourite countermodel of the de Morgan law, then remember that propositions can be seen as subsets of a singleton set in a way that maps $\cap$ to $\land$, $\cup$ to $\lor$ and $-^c$ to $\neg$. – Naïm Camille Favier Nov 04 '24 at 12:52
  • @NaïmFavier so you're suggesting a set-theoretic model "based off" of a countermodel for the De Morgan law? – Atom Nov 04 '24 at 13:32

1 Answers1

2

First, $\mathsf{ZF}$ and $\mathsf{ZFC}$ are classical theories, so they contain $\mathsf{LEM}$ as an axiom. If you want to talk about intuitionistic set theory, you should refer to the intuitionistic $\mathsf{ZF}$ or constructive $\mathsf{ZF}$, denoted by $\mathsf{IZF}$ and $\mathsf{CZF}$ respectively. I do not want to explain the difference between $\mathsf{IZF}$ and $\mathsf{CZF}$ and their historical origins; You should refer to the second section of my preprint.

Regarding the unprovable part of the de Morgan's law, it is known that it is equivalent to the weak excluded middle $\lnot P \lor\lnot\lnot P$. Of course, it does not mean constructive mathematicians do accept this rule; Rather, it is more likely that constructive systems will not accept the remaining part of de Morgan's law since it is too close to the full $\mathsf{LEM}$. Yes, it may make doing mathematics harder, but in some cases (like, Bishop-styled constructive analysis) it is not too hard.

There are bunch of models not satisfying the remaining part of the de Morgan's laws (or equivalently, the weak excluded middle). You may pick almost any intuitionistic model, like, the effective topos (a realizability topos for Kleene's first algebra, a PCA given by a universal Turing machine). If you don't like toposes, you may check McCarty's doctoral thesis in which the author presents a model of $\mathsf{IZF}$ that should be 'equivalent' to the effective topos.

Hanul Jeon
  • 28,245
  • (Currently, McCarty's thesis seems unavailable online due to the cyber attack against EThOS happened last year. Feel free to contact me if you need it.) – Hanul Jeon Nov 04 '24 at 15:38
  • For future readers: Equivalence of the said De Morgan's law and WEM is shown on ncatlab. – Atom Nov 04 '24 at 19:51
  • So the tl;dr version of the first and last paragraphs (since I am not familiar with toposes or IZF or CZF) is that there are models of the "intuitionistic set theory" (which differs from ZF or ZFC) for which the said De Morgan's law is not true (and this is mentioned in McCarty's thesis). Correct? – Atom Nov 04 '24 at 19:57
  • @Atom Yes, correct. I am unsure if McCarty clearly stated that WLEM fails in his model, but it should follow from, like the failure of WLPO over his model. – Hanul Jeon Nov 04 '24 at 21:59