If 'f(x) holds for all cardinals' simply IS the conjunction of base case and inductive step, then the infinity of arithmetic truths collapses into a finite linguistic act — an implausible consequence that Gödel's incompleteness results make formally precise.
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conjunction(Argument about atomic parthood via directional contact)
The coming together or contact of atoms from different directions, used to test whether atoms have spatial parts.
f(x)(as used in mathematical logic)
A mathematical function or formula where x is a variable—think of it like a machine where you put in a number (x) and get out a result. The 'f' is just the name of that machine.
implausible consequence(as used in philosophical argumentation)
A conclusion that seems unlikely, unreasonable, or probably false—something that suggests a theory or argument might be wrong because its results don't make sense.
inductive step(as used in mathematical induction)
The part of a mathematical proof where you show that if something is true for one case, it must also be true for the next case—like proving dominoes will keep falling if you knock one down.
linguistic act(as used in philosophy of language)
Something you do with language—like speaking, writing, or making a claim—rather than describing something about the world itself.