This isn’t easy going, but there’s a story out, with a very good write up in Quanta, on mathematical axioms, as they apply to different sizes of infinity. Key paragraph:
Their proof [about sizes of infinities], which appeared in May in the Annals of Mathematics, unites two rival axioms that have been posited as competing foundations for infinite mathematics. Aspero and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true.
For the moment, call these two supposed axioms D and E. The idea is that, before, people believed D and E separately, and now, it is thought that starting with D we can deduce E.
Assume that that’s true. Then, starting at D, and bringing in a whole host of other axioms, and necessary truths deduced from those axioms, such as are commonly found in logic and mathematical logic, E can be shown to follow from D—a shorthand way of saying it. More completely: E can be shown to follow from D and A_1, A_2, A_3, …, A_p, other associated axioms, and B_1, B_2, B_3,…B_q, necessary truths deduced from those A_i.
I don’t know the size of the entire suite of As and Bs is, but it will be substantial.
Regular readers will know an axiom, a proper axiom, is a proposition that is believed but cannot be proved. That is, it cannot be deduced from other axioms, which include truths of logic and so forth. True axioms are believed based on intuition alone.
Here’s a for-instance from the article, where you have to know “(*)” (the whole thing with parentheses) is one of the axioms mentioned above:
“With two highly productive axioms floating around, proponents of forcing faced a disturbing surplus. ‘Both the forcing axiom [Martin’s maximum] and the (*) axiom are beautiful and feel right and natural,’ Schindler said, so “which one do you choose?'”
There is nothing wrong this Schindler’s sentence; indeed, sentiment is how we judge all axioms. On how they feel, where that word is used in its intuitionist sense. All of our most important truths cannot be proved, but must be believed—on faith, if you like.
Now about these two axioms:
If the axioms contradicted each other, then adopting one would mean sacrificing the other’s nice consequences, and the judgment call might feel arbitrary. “You would have had to come up with some reasons why one of them is true and the other one is false — or maybe both should be false,” Schindler said.
Godel’s theorem lurks under this, but for us it’s not a mystery.
The curious thing for us is that sentence at the beginning, ” Aspero and Schindler showed that one of these axioms implies the other, raising the likelihood that both axioms — and all they intimate about infinity — are true.”
“Raising the likelihood” means adding probative information to the right hand size of a probability “equation”. Not all probabilities are numbers, so “equation” is used metaphorically.
Incidentally, all this has to do with continuum hypothesis. This is that there is no size of infinity between the infinity of the countables (1,2,3…) and the reals (e, π, 0.000000000001212,…). (I’m not the only one to dislike the names of these things.)
It is known the size of the reals is larger than the size of the countables. This is also put by saying the cardinality of the reals is bigger. And it was proposed that no other kinds of infinities could fit between the countables and reals. The new supposition, flowing from being able to prove one axiom from another, is that at least one other kind of infinity can be sandwiched in.
Funny thing is probability as a formal branch of math, i.e. measure theory, only uses countables and reals. There are known to be infinities larger than reals (and none smaller than countables), so