💥 Is there an infinite set that's bigger than the set of integers but smaller than the set of real numbers? Cantor guessed the answer is *no*. This guess, shown on the shirt below, is called the Continuum Hypothesis. Now it's been connected to #machine_learning!
https://t.co/lNIQzHrU4v
In 1938 Kurt Gödel showed the #Continuum_Hypothesis cannot be *disproved* using the standard axioms of set theory (the ZFC axioms).
In 1963 Paul Cohen showed the Continuum Hypothesis cannot be *proved* using these axioms!
Since the Continuum Hypothesis can neither be proved nor disproved using the standard axioms of set theory, we say it's "independent" of these axioms.
It's surprisingly useless: I've never seen an interesting question that it would settle, except itself.
But now 5 mathematicians working on machine learning have found an interesting question whose answer is "yes" if we assume there are *at most finitely many* cardinals of size between the cardinality of the integers and that of the reals, and *no* otherwise.
🔗 https://www.nature.com/articles/d41586-019-00083-3
The claim that there are at most finitely many cardinals intermediate in size between the integers and the reals is a variant of the Continuum Hypothesis, which is *also* independent of the usual axioms of set theory.
Let me call this variant Axiom Q.
There's an unknown probability measure P on some finite subset of the interval [0,1]. You get to see some number N of independent and identically distributed samples from P.
Your task: find a finite subset of [0,1] whose P-measure is at least 2/3.
Can you?
You can always succeed in doing this task if we assume Axiom Q , but you cannot if we assume the negation of Axiom Q.
So, your ability to carry out this task cannot be determined using the standard axioms of set theory!
Read the paper for details!
🧷 https://www.nature.com/articles/s42256-018-0002-3
The surprise is not that a question coming up in machine learning turns out to be independent of the standard axioms of set theory. Lots of interesting math questions are!
The surprise is that it could be settled by a variant of the Continuum Hypothesis!
🖇 https://threadreaderapp.com/thread/1083047483368890368.html
https://t.co/lNIQzHrU4v
In 1938 Kurt Gödel showed the #Continuum_Hypothesis cannot be *disproved* using the standard axioms of set theory (the ZFC axioms).
In 1963 Paul Cohen showed the Continuum Hypothesis cannot be *proved* using these axioms!
Since the Continuum Hypothesis can neither be proved nor disproved using the standard axioms of set theory, we say it's "independent" of these axioms.
It's surprisingly useless: I've never seen an interesting question that it would settle, except itself.
But now 5 mathematicians working on machine learning have found an interesting question whose answer is "yes" if we assume there are *at most finitely many* cardinals of size between the cardinality of the integers and that of the reals, and *no* otherwise.
🔗 https://www.nature.com/articles/d41586-019-00083-3
The claim that there are at most finitely many cardinals intermediate in size between the integers and the reals is a variant of the Continuum Hypothesis, which is *also* independent of the usual axioms of set theory.
Let me call this variant Axiom Q.
There's an unknown probability measure P on some finite subset of the interval [0,1]. You get to see some number N of independent and identically distributed samples from P.
Your task: find a finite subset of [0,1] whose P-measure is at least 2/3.
Can you?
You can always succeed in doing this task if we assume Axiom Q , but you cannot if we assume the negation of Axiom Q.
So, your ability to carry out this task cannot be determined using the standard axioms of set theory!
Read the paper for details!
🧷 https://www.nature.com/articles/s42256-018-0002-3
The surprise is not that a question coming up in machine learning turns out to be independent of the standard axioms of set theory. Lots of interesting math questions are!
The surprise is that it could be settled by a variant of the Continuum Hypothesis!
🖇 https://threadreaderapp.com/thread/1083047483368890368.html
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John Carlos Baez
Is there an infinite set that's bigger than the set of integers but smaller than the set of real numbers? Cantor guessed the answer is *no*. This guess, shown on the shirt below, is called the Continuum Hypothesis. Now it's been connected to... machine learning!…