Principle of Explosion

Harvey Friedman hmflogic at
Thu May 19 01:12:12 EDT 2022

I wrote

Wiki calls this the principle of explosion and I will go with that rather
than EFQ. EFQ is being discussed by Arnon and Tennant on FOM.

This is NOT the law of excluded middle or any of its inferential variants.

I had something further to say about this.

THEOREM. The emptyset is contained in every set.

Proof: Let A be any set. To show emptyset containedin A, we need to show
the following:

if x is in emptyset then x lies in A

Assume x is in emptyset. Then absurdity. Hence x lies in A. QED

This is the way mathematicians normally prove the Theorem. THis proof uses
the principle of explosion.

The use of explosion occurs also in the proof of this theorem that does not
explicitly mention the empty set.

THEOREM'. There is one and only one set that is included in every set.

How does one prove Theorem and Theorem' without use of explosion?

Harvey Friedman
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