

To Settle Infinity Dispute, a New Law of Logic - jonbaer
https://www.simonsfoundation.org/quanta/20131126-to-settle-infinity-question-a-new-law-of-logic/

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jfarmer
This article is confused and its confusion is apparent right from the first
paragraph: "In the course of exploring their universe, mathematicians have
occasionally stumbled across holes: statements that can be neither proved nor
refuted with the nine axioms, collectively called “ZFC,” that serve as the
fundamental laws of mathematics. Most mathematicians simply ignore the holes,
which lie in abstract realms with few practical or scientific ramifications.
But for the stewards of math’s logical underpinnings, their presence raises
concerns about the foundations of the entire enterprise."

Actually, it's confused from the first sentence: the job of a mathematician
isn't to explore "the universe," which is ultimately an empirical question.

Anyhow, what Gödel proved is that such "holes" — a term a mathematician would
never use — are a necessary consequence of any sufficiently powerful theory.
In a nutshell, Gödel's incompleteness theorem says that if we have an
axiomatic system powerful enough to describe the natural numbers and
arithmetic — what is called Peano arithmetic — but manageable enough for us to
enumerate the axioms then there will be theorems expressible in that axiomatic
system which are neither provable nor disprovable in that axiomatic system.
Note that they would well be provable or disprovable in some larger, more
powerful axiomatic system. Or, like Euclid's parallel postulate, we could have
a consistent mathematics whether we assume the theorem or its negation.

Even if we extend ZF(C) with axioms which allow us to prove or disprove CH,
there will still be other theorems which are unprovable. Because CH is
independent of ZF(C), the following statement is true: "If ZF(C) is is
consistent then both ZF(C)+CH and ZF(C)+not-CH are consistent." That is, if we
believe ZF(C) is consistent, we can add either CH its negation to our set of
axioms and remain consistent. This is what it means for CH to be independent
of ZF(C).

It's also worth noting that the Axiom of Choice — the (C) in ZF(C) — is
independent of ZF. This means one can consistently do math in both ZF(C) and
ZF(not-C), assuming ZF itself is consistent.

In the history of mathematics, Cantor believed CH to be true and Gödel believe
it to be false. Nowadays, most mathematicians don't worry about this stuff.
Instead, mathematicians prove statements like "In ZF(C), statement X is true
if and only if CH is true." Whether one assumes CH or not-CH has to be decided
on other grounds, e.g., aesthetics, utility, and so on.

The independence of CH from ZF(C) makes arguing for its truth or falsity a
fool's errand. I think the error is in believing that every statement in
mathematics must be a statement about our universe, i.e., either CH is true or
it isn't and that's that. It's not a statement about our universe, however —
it's a statement about some axiomatic system we've developed. If that seems
weird to you, consider that you play with utterly un-physical constructs all
the time, like numbers and perfect circles.

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dmfdmf
Some of us just happened to be discussing this topic here
[https://news.ycombinator.com/item?id=6844885](https://news.ycombinator.com/item?id=6844885)

