Is it possible to have distinct implications from the existence of only one axiom?

jambaugh
Gold Member
Technically all axioms can be conjoined into a single postulate:

A = A1 and A2 and A3....

So every axiomatic system can be though of as having 1 axiom and the answer to your question is "Yes".

I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?

chiro
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?

Wouldn't the one axiom simply encode all the information in a way like jambaugh has said? The definition through use of intersection is universal, it doesn't take context depending on the axiom or the system/constraints its describing.

jambaugh
Gold Member
I know what you mean, but wouldn't you need an axiom that allows you to "combine" the axioms into one logical statement.

Anywho let me be more specific to dodge your problem then, assume you have only one axiom, the axiom of extensionality from ZFC. Can any truly distinct implications be concluded from this axiom?

Again this depends on what you mean (I think your question is ill posed).

Suppose you have a system of axioms A1, A2, and A3 from which you formulate a set of definitions and prove a theorem T.

From just A1 you can prove T' = (A2 and A3 implies T).

By the same token you can start with 0 axioms and change each theorem to the corresponding contingent theorem. e.g. T'' = (A1 and A2 and A3 implies T).

Unless you get very specific about the format of theorems and axioms, counting how many you start with is not very meaningful. The math is not in the axioms per se but in the implication structure.