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60 lines
3.2 KiB
Markdown
60 lines
3.2 KiB
Markdown
---
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lastmod: "2021-09-12T22:47:44.0000000+01:00"
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author: patrick
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categories:
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- mathematical_summary
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comments: true
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date: "2016-04-13T00:00:00Z"
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math: true
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aliases:
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- /independence-of-choice/
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title: Independence of the Axiom of Choice (for programmers)
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summary: So you've heard that the Axiom of Choice is magical and special and unprovable and independent of set theory, and you're here to work out what that means.
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---
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So you've heard that the Axiom of Choice is magical and special and unprovable and independent of set theory,
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and you're here to work out what that means.
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Let's not get too hung up on what the Axiom of Choice (or "AC") actually is, because you probably don't care.
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Let's instead discuss what it means for something to be "independent".
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Often I hear the layperson say things like "AC is unprovable".
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This is true in a sense, but it's misleading.
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Take an object \\(n\\) of the type "integer" - so \\(5\\), \\(-100\\), that kind of thing.
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Here is what I will call the Positivity Hypothesis (or "PH"):
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> \\(n\\) is (strictly) greater than \\(0\\).
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Of course, depending on how we chose \\(n\\), PH might be true or it might be false, although it can't be both.
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So, while maths might let us prove which of PH or not-PH holds for our given \\(n\\),
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maths will emphatically not let us prove that PH is always true, and it will not let us prove that PH is always false.
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(Maths would be stupid if it did that, because PH is neither always true nor always false.
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The integers \\(5\\) and \\(-100\\) witness that PH can be true and can be false respectively.)
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Therefore PH is independent of integer theory.
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It's not magic: there is no god-given reason why PH mysteriously resists all efforts to prove it.
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It's simply not always true, but it's not always false either.
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Let's go back to the Axiom of Choice.
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The usual system of set theory (which is used as a foundation for all of maths) is a collection of nine axioms,
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together comprising what is known as ZF.
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(If we add Choice to that collection as a tenth axiom, we obtain the set theory called ZFC.)
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In the "integers" analogy above, "the integer type" plays the role of ZF.
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Now, just as we may pick an object of type "integer", we may pick a set-theory of type "ZF".
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A "set theory of type ZF" is my informal phrasing for what is usually called "a model of ZF".
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(I'm eliding the question of the consistency of ZF, and I'll just assume it's consistent.)
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In the "integers" analogy, the number \\(5\\) plays the role of one of these set theories,
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as does the number \\(-100\\).
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We can ask of this set theory whether it obeys AC (for which we substituted PH in the analogy).
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And it turns out that for some models of set theory, AC holds, and for some models, it doesn't.
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It's quite hard to describe models of set theory, because set theory supports so much complexity;
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the integers are much easier to specify.
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However, if you want the names of two models: in the model which contains precisely the "constructible sets", AC holds, while in Solovay's model, AC fails.
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That's all there is to it.
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Maths won't let us prove AC, because it's not true of every set theory of the type "ZF".
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Maths won't let us prove AC is false, because there are some set theories of the type "ZF" in which it is true.
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