Principle of Explosion

Tennant, Neil tennant.9 at osu.edu
Fri May 20 19:33:57 EDT 2022

```Harvey,  you asked:

"How does one prove Theorem and Theorem' [easy results about the empty set--NT] without use of explosion?"

The answer is: Very easily in Core Logic, as explained in my book Core Logic, Chapter 12, 'Replies to Critics of Core Logic', Section 5: 'Reply to Friedman and Avron'; pp.327-337.

It is a very nice rebuttal of you and Avron, but too long to fit into the margins of this note to fom.  🙂

Best regards,
Neil

________________________________
From: FOM <fom-bounces at cs.nyu.edu> on behalf of Harvey Friedman <hmflogic at gmail.com>
Sent: Thursday, May 19, 2022 1:12 AM
To: Foundations of Mathematics <fom at cs.nyu.edu>
Subject: Principle of Explosion

I wrote https://cs.nyu.edu/pipermail/fom/2022-May/023297.html  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

I wrote https://cs.nyu.edu/pipermail/fom/2022-May/023297.html<https://urldefense.com/v3/__https://cs.nyu.edu/pipermail/fom/2022-May/023297.html__;!!KGKeukY!zRljlSLX5_TL63wtFM2YxcpIQ8etGiBTJfUefGcimqVSfV3YbHy9Ajk6KZ3StY9J7OFNnYyijL7izd6LZA\$>

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.

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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