It would seem superficially this is correct

A=A, necessarily and infallibly true

but it is false for non-trivial A and where the EQUAL SIGN means equal in essence, not just equal in description. This is a consequence of the dictionary and halting problem that has been well demonstrated by first rate mathematical logicians and computer scientists (like Alonzo Church who was a devout Christian) who actually work with formal languages vs. people clinging to antiquated and non-rigorous theological notions.

Now consider this

Property (4 quarters) = Property (100 pennies)

Depending on how Property operator works, the above is could be true or it could be false.

Similarly we are posed with the same issue here

Property(1+1) = Property(2)

false under modulo 2 arithmetic, true under Peano and Real axioms.

One could construct a moral code whereby…

Property(eating oranges) = KOSHER

Property(eating pork) = NOT KOSHER

or a moral code whereby

Property(eating oranges) = KOSHER

Property(eating pork) = KOSHER

The issue is you can make symbols mean almost whatever you want! So this fact isn’t reassuring for those demanding infallibility.

One work around is to prevent any sort of meaning. So then we allow only identical symbolic strings to be declared as equal. Hence under such a regimen:

“1+1″=”2”

is false since the literal strings are not identical! But such a regimen isn’t very useful. We want to have a system the EQUAL SIGN allows for equality of essence independent of description.

Shakespeare captured the idea there is an essence that transcends description:

A rose is a rose by any other name

But in the world of symbols, the description is the essence. We could create a substitutional machinery that says

1+1 = 2 = 3-1= 5-3 = 1+1-1+5-3-3+2=…..

and thus A=A if we are talking about A’s essence, not its description.

The problem however is if we have a dictionary of symbols, and we can create meaning by substituting all the symbols with other symbols, we have circularity. Nothing becomes clearer by trying to define symbols by other symbols in the same dictionary. In the search for meaning, we end up with a halting problem!.

We try to say the EQUAL SIGN means EQUAL IN ESSENCE but not in description. We could then demand the EQUAL SIGN means only equal in description, but then one has a trivially useless meaning for the EQUAL SIGN.

So if we allow the equal sign to permit substitutions and meaning, and say the EQUAL SIGN refers to equality in essence, not description, then we can say things like this in the axioms of Peano or the Real Numbers:

1+1 = 2

4+3= 7

etc.

But Godel demonstrated, there will be some things of the same essence but different descriptions which one cannot prove as having the same essence. That is, you will not be able to prove A=A even if true for some A in terms of essence because of the halting problem. It is a true but an undecidable proposition, hence it is true but not necessarily true, hence this statement is false

A=A, necessarily and infallibly true

for non-trivial A where the EQUAL SIGN means equal in essence, not equal in symbolic description.

Quod Erat Demonstrandum.

No we don’t. Describe the halting problem in your own words and explain how it applies here.

The usual view within mathematics, is that “=” means identical.

So 1+1=2

is understood to say that “1+1” and “2” are two different names for the exact same entity (the number one which some folk take to be a platonic entity and others take to be a useful fiction).

However “equal” in ordinary life means something different, as for example “all men are created equal”.

That’s a creative way of describing Gödel’s result. It’s so creative, that I’m having trouble making sense of it.

Sal,

You’re confusing symbolic representations with the numbers being represented.

Property(2) is meaningless in modulo-2 arithmetic because 2 doesn’t exist. In modulo-2 arithmetic, 1 + 1 is identical to 0, so any property possessed by 1+1 is also possessed by 0.

In modulo-3 (or higher) arithmetic, any property possessed by 1+1 is also possessed by 2, since both represent the same underlying number.

Consider:

1.0 = .999999999999 repeating

Two different representations of the same thing.

https://en.wikipedia.org/wiki/0.999…

If we have a non-computable number with two different representations, we won’t be able to prove them to be identical even though they are.

If they hypothetically have the same representation, no problem, but that’s trivial. Like saying:

“1+1” = “1+1”

But they cannot be represented in a formal system, so it’s not provable anyway.

Assume a system obeys FieldAxioms:

http://mathworld.wolfram.com/FieldAxioms.html

SquareRoot(A) = SquareRoot(A)

True if the field are the Reals, meaningless if the field are the Rationals.

If a non-computable number has no representation in a system, it’s sort of moot to be saying it can be compared since the comparison is undecidable.

https://en.wiktionary.org/wiki/noncomputable

Like saying

Smallest Fraction = Smallest Fraction

When StephenB says “same ontological sense”, that’s non-rigorous philosophical mush, certainly doesn’t have much utility in symbolic representation.

But how does that relate to Gödel? His proof used only integers, and depended on the unique factorization into products of primes.

Under the axioms of the reals, even for non-computable numbers:

A + (-A) = 0

But that is an axiom, not a proposition, so it is defined as true by assumption, that is different than true by necessity.

Keiths,

You are correct, my mistake. Thanks for pointing it out. I should have said:

Property(1+1) = Property(0)

under modulo 2.

First off, I’m not disputing A=A in reals as an axiom, I’m disputing it as a proposition. It is true by assumption, not by necessity.

Would you prefer Tarski’s undefinability of TRUTH theorem?

http://www.math.hawaii.edu/~dale/godel/godel.html

I’m just trying to say, A=A can be accepted as true as an axiom, not as a proposition.

No offense, Sal, but you’re so confused that it would take hours trying to sort it out. For one thing, you’re mixing up two different senses of the word “undecidable”. Godelian undecidability is not the same as “undecidable” in computation theory (which is nowadays more usually called “uncomputable”).

See, for example, https://en.wikipedia.org/wiki/Undecidable_problem and read the paragraph starting “There are two distinct senses of the word “undecidable” in contemporary use. “

Jeff!

Hey thanks for the criticism! No offense taken. Thanks for reading.

Sal

I admit I was lost reading that OP. So I too thank you.

Aren’t Dembski and Marks the greatest though!? Cant’ wait for the second edition of

NFL.Jeff,

I know you are a recognized expert on computational number theory. I’m indebted to you for pointing out my misunderstanding. Thank you for the wiki link:

Just having one of my random-with-respect-to-function post purges. Some posts moved to Guano.

Please try to stick to the rules guys

ta.

In light of Jeff Shallit and Niel’s criticisms, this sentence must be withdrawn and reworded since as Jeff pointed out there two notions of Undecidable and one sense of the word applies to computation number theory.

Proposed rewording along the following lines:

1+1 = 2

What we really mean is something like

A = Property(1+1) = Property (2) = A

which reduces to

A =A

that is because the meaning of the EQUAL SIGN goes deeper than mere identical description, but some substitution of symbols, i.e. we can substitute the string “1+1” with the string “2” under our rules.

What is not under dispute in the real numbers is that A=A is pretty much an axiom. I claim it cannot be offered up as a proposition. A=A is true by assumption, no necessity that can hypothetically be proven or disproven.

If we have non-computable numbers, their representation is undecidable. The reals have far more non-computable numbers than computable numbers. If we have say the following numbers, B, C, D,E:

A= B+C = D+E = A

that hypothetically could reduce to

A=A

if we could actually compute the value, but we actually can’t because A is non-computable, then we have a case where A=A is true but cannot be proven as such.

There will be some A=A that we can only assert by assumption, not by necessity.

The new version is a big improvement.

I have limited keyboard access over the next few days, so I won’t give detailed comments.