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lavalamp comments on Standard and Nonstandard Numbers - Less Wrong
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Thanks. Hm. I think I see why that can't be said in first order logic.
...my brain is shouting "if I start at 0 and count up I'll never reach a nonstandard number, therefore they don't exist" at me so loudly that it's very difficult to restrict my thoughts to only first-order ones.
This is largely a matter of keeping track of the distinction between "first order logic: the mathematical construct" and "first order logic: the form of reasoning I sometimes use when thinking about math". The former is an idealized model of the latter, but they are distinct and belong in distinct mental buckets.
It may help to write a proof checker for first order logic. Or alternatively, if you are able to read higher math, study some mathematical logic/model theory.