

Set Theory and Axiomatic Systems - archibaldJ
http://0a.io/0a-explains-set-theory-and-axiomatic-systems-with-pics-and-gifs

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tunesmith
Interesting that the first axiom he highlights is considered rather
controversial. I only have a layman's understanding, but my sense is that the
answer to "(If A is true) Do you believe not-not-A is true?" points to whether
you subscribe more to classical logic or intuitionist/constructive logic.

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cottonseed
It's double-negative elimination that does hold constructively, namely, not-
not-A => A. In both classical and constructive logic, A implies not-not-A.

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PeterWhittaker
The double-negation holds in propositional logic (law of the excluded middle)
but definitely not in constructive logic: You don't get that for free, you
need a path from ~A to ~~A to A.

(And, refining the logical system to include bounds due to computational
complexity, you need a path that is realistic in finite time.)

It also doesn't apply in Hegelian logic: ~A is literally everything that isn't
A - and negating is likely undefined, since there is nothing in the universe
that makes it necessary (regardless of sufficiency) for ~EverythingElse to
necessarily mean that anything is left over, let along something as specific
as A.

Unless A is somehow unspecific, in which case even ~A is likely defined and
potentially meaningless.

I think the two worst intellectual holdovers in Western thought are
Aristotelianism - and especially the law of the excluded middle - and
Descartes' broader thinking - and especially his duality and "Cogito Ergo
Sum": Both were really useful as developments in epistemology and logic, but
both have outlived their then-value. We have moved on.

They are not even particularly useful as logical pablum for children, because
it is too easy to assume forever that they are 100% reliable and valid - they
are useful rules of thumb, they apply in many common, real world cases, but
they are, logically, nothing more than extremely useful heuristics.

One can build vast systems with them, but one eventually hits Gödel and
Aaronson, e.g., and one realizes the systems are built on less than sand.

Once one gets past them, one can begin to grok where mathematics and physics
are now. But they are hard to break away from.

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cottonseed
I'm a mathematician. I can't understand your post at all.

> The double-negation holds in propositional logic

Double-negation isn't a proposition, so don't know what this means. Double-
negation elimination, that is, the proposition: for all A, ~ (~ A) -> A does
not hold in constructive logic.

However, double-negation introduction, that is, the proposition: for all A, A
-> ~(~ A) does definitely hold in constructive logic. Here is a proof in Coq:

Theorem A_implies_not_not_A : forall A : Prop, A -> ~ (~ A). unfold not.
intros. apply (H0 H). Qed.

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tromp
A small error I noticed:

After explaining the difference between subset and strict_subset, they claim
that for all A

empty_set strict_subset A

which obviously fails when A is itself empty.

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archibaldJ
Nice! I was just waiting for someone to point it out.

