Possible consequences of falsifying the principle of explosion?

[wtf]

I'd like to make some comments on your reply. I don't feel that I"m arguing a position, so you don't need to snap back at each point.

I'm just outlining some of the obvious objections that:

1) Will be asked by ANYONE you present your ideas to;

2) And that in my opinion, you haven't handled very well.

You say you are trying to get your ideas understood. I hope you'll take my comments as constructive criticism and pointers to things you should consider.

The axiom of choice seems obviously correct.

Yes of course. It does seem obviously correct, so much so that we are surprised to find that it does not follow from the other axioms of ZF. It also has many strange consquences, of which the famous Banach-Tarski paradox is perhaps the most striking. For that reason, many people dislike it.

In 1938, Gödel showed that if ZF is consistent in the first place, then ZFC -- ZF with the axiom of choice (AC) added -- is also consistent. He did this by exhibiting a model of ZF in which AC is true.

I know you don't like models, but you should try to come to terms with the concept. They're essential to mathematical logic.

Then in 1963, Paul Cohen produced a model of ZF in which AC is false.

This shows that the axioms of ZF do not decide AC one way or the other; and that it's perfectly consisten to either assume AC or its negation.

You say you don't want to have to learn all the "details of math," but in fact you should consider trying to understand this particular example. It is the classic example of an undecidable statement

You can, in your own terminology, make a "stipulative definition" that AC is true, and you can do math. Or you can stipulate that it's false, and do math.

But you claim that every closed wff has a definite truth value. Therefore you are making the claim that your system can show what is the correct truth value of AC.

For the sake of anyone ever taking your ideas seriously, you have got to come to terms with this example. Nobody knows if AC is true or false, or even if the question is meaningful. After all it's about abstract, infinite sets. It's hard to imagine that it even has a truth value. Abstract, infinite sets have no referents in the real world as far as we know. It's like asking whether Captain Ahab likes eggs for breakfast. He's a fictional character and the novel Moby Dick doesn't say anything about Ahab's breakfast. There is no truth value to be had.

On the other hand, Platonists do feel that AC has a definite truth value, if we could only find the "right" axioms. So there's a lot of philosophy and a lot of mathematical logic here, and it's something you should grapple with.

I spent a very long time carefully composing a reply and the system erased it.

I'm composing this in file in a text editor. When I'm done I'll paste it into the forum and fix up any typos. If you do that you won't lose posts to forum hiccups.

What I am stating is the whole idea of stipulated definitions are irrefutable.

We know that we can stipulate AC or its negation, and that we have no objective way of knowing which is true. AC can not in fact be proved from ZF nor can the negation of AC.

You have to deal with this. You have to at least try to understand it. You say that you can stipulate what you like, and you are correct. But just randomly stipulating things can never lead you to truth.

When 2 + 3 = 5 is defined to be true, you can't say wait wait I have a counter-example.

But I did not say anything about 2 + 3 = 5. That is easy to prove in ZF.

I'm talking about AC, which is provably independent of ZF. And now I CAN say wait, here is a counterexample.

This is semantic tautology, not quite the same thing as logical tautology.

https://www.britannica.com/topic/tautology
Tautology, in logic, a statement so framed that it cannot be denied
without inconsistency.

We are not talking about any tautologies.

Thus, “All humans are mammals”
prevents any rebuttal that humans are not mammals.

Pete, that is NOT a tautology. A tautology is a closed wff that is true under every interpretation of its symbols. But "all humans are mammals" is only true when humand and mammals are given their usual interpretation. There is nothing structurally necessary about the statement. It is NOT a tautology.

Please put aside some time in your life to understand what a tautology is. Any statement that depends for its truth value on the MEANING of its symbols, is NOT a tautology.

A tautology is true solely by virtue of its structure and never its meaning.

Combining the conventional notion of {formal proofs of mathematical logic}
with the conventional notion of {sound deductive inference} necessarily
creates {sound deductive formal proofs of mathematical logic} which does
indeed necessarily reject some expressions of language as deductively unsound.
All of this is a matter of definition, thus not subject to any counter-examples.

This is just some word salad that you revert to whenever you want to avoid dealing with an issue. Nobody is fooled, you're only fooling yourself.

Any possible counter-example to the contrary would be exactly the same thing
as arguing against any stipulated definition such as: “All humans are mammals”.

Do you stipulate AC or its negation? Both are equally consistent with ZF and neither can be proven from ZF.


Now my second point. You have been consistently ignoring my question about your redefinition of material implication. You said that you define material implication as having the same truth table as logical conjunction, or AND.

I asked an obvious question. Given "2 + 2 = 4 AND Washington is such and so," and "2 + 2 = 4 THEREFORE Washington such and so," are you claiming that these two statements have the same meaning?

That's the consequence of your assiginng AND and THEREFORE the same truth table

I've presented this example to you four times now, and each time you simply ignore it. Pete, it's not my problem. You're the one who wants to be understood. If you don't want to grapple with the question, you don't have to. But nobody will take you seriously if you keep avoiding questions.

All the best. I have done what I can to point out some weak areas in your presentation. The rest is up to you.

[PeteOlcott]

Do you stipulate AC or its negation? Both are equally consistent with ZF and neither can be proven from ZF.

I would rather not open the can-of-worms of set theory.
The people specifying these things have proven their incompetence.

Even after all of these decades they still did not even solve Russell's paradox correctly.
To solve Russell's paradox correctly only requires one to understand that no physical or
conceptual thing can ever totally contain itself.

∀x ∈ Thing (¬Totally_Contains(x, x))

Imagine a can of soup totally containing itself such that it has no outside boundary.
Now draw a Venn diagram of that.

[PeteOlcott]

This is semantic tautology, not quite the same thing as logical tautology.

https://www.britannica.com/topic/tautology
Tautology, in logic, a statement so framed that it cannot be denied
without inconsistency.

We are not talking about any tautologies.

Thus, “All humans are mammals”
prevents any rebuttal that humans are not mammals.

Pete, that is NOT a tautology. A tautology is a closed wff that is true under every interpretation of its symbols. But "all humans are mammals" is only true when humand and mammals are given their usual interpretation. There is nothing structurally necessary about the statement. It is NOT a tautology.

We are talking about SEMANTIC TAUTOLOGIES, see the link.
my example is a quote for the link.

This is not exactly that same thing as a logical tautology.
It is more like a stipulated definition.

When it is stipulated that: "All humans are mammals" any counter-example
is defined to be logically incorrect.

The actual nature of truth itself it just an interconnected set of stipulated definitions.
When the conventional definition of the ordered set of integers is stipulated then
5 > 3 is specified. Counter-examples to stipulated definitions are not allowed.

[wtf]

no physical or
conceptual thing can ever totally contain itself.

See https://en.wikipedia.org/wiki/Non-well- ... set_theory

Counter-examples to stipulated definitions are not allowed.

Are you honestly under the impression that this is a coherent response to what I wrote?

[PeteOlcott]

no physical or
conceptual thing can ever totally contain itself.

See https://en.wikipedia.org/wiki/Non-well- ... set_theory

Counter-examples to stipulated definitions are not allowed.

Are you honestly under the impression that this is a coherent response to what I wrote?

I barely read what you wrote. My point does boil down to this though:
Within the correctly combined semantic meaning of conventional
{formal proofs of mathematical logic} and conventional {sound deductive inference}
every undecidable sentence of conventional {formal proofs of mathematical logic}
is deductively unsound.

This is true in the same way that 5 > 3 is true, it is a semantic tautology.

[wtf]

I barely read what you wrote.

All the best brother.

[PeteOlcott]

I barely read what you wrote.

All the best brother.

Everything that anyone could ever possibly say is already addressed
by the concept of semantic tautology that I specified in my prior reply.
I don't have time to infinitely digress from this its already been 22 years.

[PeteOlcott]

no physical or
conceptual thing can ever totally contain itself.

See https://en.wikipedia.org/wiki/Non-well- ... set_theory

Counter-examples to stipulated definitions are not allowed.

Are you honestly under the impression that this is a coherent response to what I wrote?

When I say something analogous to: ¬∃x ∈ Integers (x > 5 ∧ x < 5)
Can you see that any suggestion that I look over here, or over there, can only be fruitless?

Instead of looking for such an Integer in the Axiom of Choice or anywhere else
wouldn't it make more sense to cut-to-the chase and categorically prove that
such an integer cannot possibly exist?

If we don't do this then there is a never ending list of places that we can look for
something that is categorically impossible.

[wtf]

Instead of looking for such an Integer in the Axiom of Choice ...

That's pathetically incoherent.

[PeteOlcott]

Instead of looking for such an Integer in the Axiom of Choice ...

That's pathetically incoherent.

The concept that it is fruitless to look for an integer that is both greater than and less than 5 is incoherent?

No matter where we look for a categorically impossible thing whether we look in a coherent place or an incoherent place our search will be necessarily fruitless because it is a categorically impossible thing.

[wtf]

Instead of looking for such an Integer in the Axiom of Choice ...

That's pathetically incoherent.

The concept that it is fruitless to look for an integer that is both greater than and less than 5 is incoherent?

What do you think that would have to do with the axiom of choice? I really want to hear this.

How can it be that in 22 years you never bothered to learn about the very obvious counterexamples to your claims?

[PeteOlcott]

That's pathetically incoherent.

The concept that it is fruitless to look for an integer that is both greater than and less than 5 is incoherent?

What do you think that would have to do with the axiom of choice? I really want to hear this.

How can it be that in 22 years you never bothered to learn about the very obvious counterexamples to your claims?

There is lots of gibberish in set theory. I don't have the patience to untangle it.
If you sufficiently understand formal proofs and you sufficiently understand sound deduction
then you already understand that the intersection of these two by itself specifies
a necessarily complete and consistent formal system.

From true premises and valid inference a true consequence necessarily follows.
There are no loopholes or possible exceptions.

How would you plug your "obvious counterexamples" into that?