Conditional on the statement being meaningful, infinitude of the universe doesn't affect the statement's truth. If the meaningfulness is in question then I'm confused so wouldn't assign very high or low probabilities to anything.

Essentially:

• I have a very strong intuition that there is a unique (up to isomorphism) mathematical structure called the "non-negative integers"
• I have a weaker intuition that statements in second-order logic have a unique meaningful interpretation
• I have a strong intuition that model semantics of first-order logic is meaningful
• I have a very strong intuition that the universe is real in some sense

It's possible that my intuition might be wrong though. I can picture the integers in my mind but my picture isn't completely accurate - they basically come out as a line of dots with a "going on forever" concept at the end. I can carry on pulling dots out of the "going on forever", but I can't ever pull all of them out because there isn't room in my mind.

Any attempt to capture the integers in first-order logic will permit nonstandard models. From the vantage point of ZF set theory there is a single "standard" model, but I'm not sure this helps - there are just nonstandard models of set theory instead. Similarly I'm not sure second-order logic helps as you pretty much need set theory to define its semantics.

So if I'm questioning everything it seems I should at least be open to the idea of there being no single model of the integers which can be said to be "right" in a non-arbitrary way. I'd want to question first order logic too, but it's hard to come up with a weaker (or different) system that's both rigorous and actually useful for anything.

I've realized one thing though (based on this conversation) - if the universe is infinite, defining the integers in terms of the real world isn't obviously the right thing to do, as the real world may be following one of the nonstandard models of the integers. Updating in favor of meaningfulness not being dependent on infinitude of universe.