All these articles are difficult for me to follow, but I get the feeling that they are approximately about this:
it is difficult to define precisely (using some specific set of rules) what exactly is a “natural number”
such definitions typically include all the real natural numbers, but allow possibility of some weird numbers, too
the weird numbers make many kinds of proofs impossible, because while a theorem may be true for all truly natural numbers, it may be false for some weird numbers and true for other weird numbers, therefore impossible to prove (using some specific set of rules)
you can remove some of the weird numbers, but never all of them with a single axiom; to remove all of them you would need infinitely many axioms, which is forbidden
And the articles like this explore the various consequences of “if you are allowed to have finitely many axioms of given type, you cannot really precisely define natural numbers”, and find many things that fall apart because of that.
You can define the natural numbers using certain rules (the Peano axioms). These rules make reference to sets, so they are in turn governed by the rules of set theory (ZFC), which admit several models.
What the paper goes on to talk about is that even if you just use the natural numbers—not some weird system—then there will still be some statements whose truth depends on the ambient model of set theory. In other words Model 1 and Model 2 of ZFC can both admit the natural numbers, but they can still disagree about what’s true. Certain things will always be true, but others will be true or false depending on which model you use.
I think so. I’m confused as to whether these results can apply to the actual standard natural numbers, or just partly-weird sets which some model of set theory believes are the standard numbers—but I think it’s the second one. The paper itself says that “satisfaction is absolute,”
whenever the formula φ has standard-finite length in the meta-theory (which is probably closer to what was actually meant by those asserting it)
which appears to mean that none of the statements being disagreed about are actual mathematical statements, but rather “weird numbers” that the models think encode statements. (Note that any other way of defining mathematical formulas would probably be at least as ambiguous as the natural numbers.)
My understanding passes the first test which occurs to me, namely, it would not allow a model of set theory to realize that it has the wrong natural numbers. (If my interpretation predicted that a model with weird numbers could look at another model which disagrees about arithmetical truth and realize, ‘Wait, these disagreements only arise when we allow weird numbers, so my numbers must be partly weird,’ then I would have to be wrong.) Evidently if even one of two models can look at the satisfaction relation that the other uses to define arithmetical truth (for example), then they must agree.
All these articles are difficult for me to follow, but I get the feeling that they are approximately about this:
it is difficult to define precisely (using some specific set of rules) what exactly is a “natural number”
such definitions typically include all the real natural numbers, but allow possibility of some weird numbers, too
the weird numbers make many kinds of proofs impossible, because while a theorem may be true for all truly natural numbers, it may be false for some weird numbers and true for other weird numbers, therefore impossible to prove (using some specific set of rules)
you can remove some of the weird numbers, but never all of them with a single axiom; to remove all of them you would need infinitely many axioms, which is forbidden
And the articles like this explore the various consequences of “if you are allowed to have finitely many axioms of given type, you cannot really precisely define natural numbers”, and find many things that fall apart because of that.
Am I at least approximately correct here?
Here’s what I gather:
You can define the natural numbers using certain rules (the Peano axioms). These rules make reference to sets, so they are in turn governed by the rules of set theory (ZFC), which admit several models.
What the paper goes on to talk about is that even if you just use the natural numbers—not some weird system—then there will still be some statements whose truth depends on the ambient model of set theory. In other words Model 1 and Model 2 of ZFC can both admit the natural numbers, but they can still disagree about what’s true. Certain things will always be true, but others will be true or false depending on which model you use.
I think so. I’m confused as to whether these results can apply to the actual standard natural numbers, or just partly-weird sets which some model of set theory believes are the standard numbers—but I think it’s the second one. The paper itself says that “satisfaction is absolute,”
which appears to mean that none of the statements being disagreed about are actual mathematical statements, but rather “weird numbers” that the models think encode statements. (Note that any other way of defining mathematical formulas would probably be at least as ambiguous as the natural numbers.)
My understanding passes the first test which occurs to me, namely, it would not allow a model of set theory to realize that it has the wrong natural numbers. (If my interpretation predicted that a model with weird numbers could look at another model which disagrees about arithmetical truth and realize, ‘Wait, these disagreements only arise when we allow weird numbers, so my numbers must be partly weird,’ then I would have to be wrong.) Evidently if even one of two models can look at the satisfaction relation that the other uses to define arithmetical truth (for example), then they must agree.