I also agree, that the author probably not meant to establish an axiom: The axiom being established, while not having any support right now, does seem like something we can reduce in the future. The author also uses the word "currently" in their axiom, which contradicts axioms (or is temporal axioms a thing?).
I think the author merely meant to establish the scene for the article. Something I truly appreciate.
"unprovability" is not a property that it is necessary to prove to pick something as an axiom.
There is generally a project to reduce axioms to the simplest and weakest forms required to make a proof. This is does result in axioms that are unprovable but does not mean the "unprovable" is a necessary property of axioms.
While practice, axioms are often statements that we all agree on and accept as true, that isn't necessarily true and isn't the core of it's meaning.
Axioms are something we postulate as true, without providing an argument for its truth, for the purposes of making an argument.
In this case, the assertion isn't really used as part of a argument, but to bootstrap an explanation of how words are represented in LLMs.
Edit: I find this so amusing because it is an example of learning a word without understanding it.