Since non-standard natural numbers come after standard natural numbers, I will also have noticed that I’ve already lived for an infinite amount of time, so I’ll know something fishy is going on.
The problem is that nonstandard numbers behave like standard numbers from the inside.
Nonstandard numbers still have decimal representations, just the number of digits is nonstandard. They have prime factors, and some of them are prime.
We can look at it from the outside and say that its infinite, but from within, they behave just like very large finite numbers. In fact there is no formula in first order arithmatic, with 1 free variable, that is true on all standard numbers, and false on all nonstandard numbers.
You have the Turing machine next to you, you have seen it halt. What you are unsure about is if the current time is standard or non-standard.
Since non-standard natural numbers come after standard natural numbers, I will also have noticed that I’ve already lived for an infinite amount of time, so I’ll know something fishy is going on.
The problem is that nonstandard numbers behave like standard numbers from the inside.
Nonstandard numbers still have decimal representations, just the number of digits is nonstandard. They have prime factors, and some of them are prime.
We can look at it from the outside and say that its infinite, but from within, they behave just like very large finite numbers. In fact there is no formula in first order arithmatic, with 1 free variable, that is true on all standard numbers, and false on all nonstandard numbers.
In what sense is a disconnected number line “after” the one with the zero on it?
In the sense that every nonstandard natural number is greater than every standard natural number.