"....before the creation of these axioms, and the subsequent generation of small numbers as theorems, didn’t the small numbers already exist?'

A husband and wife are checking out of a hotel.

Receptionist: "That will be $400, sir"

Husband: "But we only stayed one night! Why is it so expensive???"

Receptionist: "We are a 5 star hotel, sir, with a world class private golf course and one of the finest spas in the country."

Husband: "But we didn't use the golf course, and we didn't go to the spa!"

Receptionist: "I understand, sir, but it was there if you wanted to."

The husband pulls out $100 and hands it to the receptionist.

Receptionist: "excuse me, sir, but you're $300 short"

Husband: "The cost for sleeping with my wife is $300"

Receptionist: "I did not sleep with your wife!!!"

Husband: "I understand, but she was there if you wanted to."

"Existence" and "availability" aren't necessarily the same thing. I'm not sure you have to say that the small numbers "existed" in order to say that they were always "available."