Excellent candidate for the label "axiom"? If 0/0 isn't equal to 25, then 0 is equal to 0.

[Boojatta]

Introduction to question:
A model-theoretic interpretation for the language used in some statement allows us to assign a value of true or false to the statement.

Question:
Is there some object analogous to a model-theoretic interpretation that allows us to determine that, relative to that object, a given formula is well-formed, or that allows us to determine that, relative to that object, the given formula isn’t well-formed?

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Link to a related thread

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Long-winded explanation of the title of this thread:
The title of this thread is intended to appeal to people who say that it is a sign of ignorance to speak of an axiom as being true, and that it's a sign of ignorance to speak of an axiom as being false.

For example, consider the following statement formulated in ordinary English:
#1. A non-zero factor that appears in both the numerator and denominator of a fraction can be canceled.

Or consider an equivalent statement formulated using mathematical symbols:
#2 For every b, and every c, if b isn't equal to zero, then (cb)/b = c.

Many people are predisposed to assert that the statements labeled #1 and #2 above are both true statements. Thus, any person who also asserts that the statement labeled #2 is an axiom faces the risk of being classified as an ignorant person. On the other hand, many people are predisposed to assert that "If 0/0 isn't equal to 25, then 0 is equal to 0" is neither true nor false, but is meaningless. Thus, in contrast with the statement labeled #2, the statement "If 0/0 isn't equal to 25, then 0 is equal to 0" can be said to be an axiom without creating excessive risk that a person who says that it's an axiom will be classified as an ignorant person.

[sailor65]

1. 42 n/t

[nebenaube]

2. question...

Is this for a math proof or a statement parsing classifier?

[Boojatta]

3. It's for a mathematical proof, or for many mathematical proofs.

It's not about computer software.

[Jim__]

4. In mathematics, For all x, x = x is already an axiom.

The axiom that you are proposing puts an unnecessary constraint on an already existing axiom - for instance see: http://en.wikipedia.org/wiki/Axiom#Propositional_logic

[Boojatta]

6. What about the option of taking "for all x, if x isn't equal to zero, then x = x" to be an axiom?

If we can prove that there exists some number (say "n") that isn't equal to 25, then -- imitating the reasoning used in the thread linked from the Original Post of this thread -- we can derive the conclusion that if 0/0 isn't equal to n, then 0 = 0.

If 0/0 were equal to 25 and 0/0 were also equal to n, then by transitivity of equality we would conclude that 25 is equal to n, but we assumed at the beginning that n was known to be unequal to 25. Thus, we can conclude that either 0/0 isn't equal to 25 or 0/0 isn't equal to n.
We have a situation of the form "if p then r" and "if q then r" and "p or q", so we are entitled to conclude that r is true. In other words, we have derived 0 = 0 as a theorem, so we have the option of weakening our axiom of reflexivity. Why use a strong assumption when we can weaken the assumption and then get back the original strength by means of reasoning rather than mere postulation?

[dimbear]

10. Try this.

You say x = x.

Then = = =.

Furthermore, = = = = =.

Let your eyes drift off center. You have to agree that

one '=' is the same as three '=' if what you claim as an axiom is an axiom.

[Jim__]

11. Read the article linked in post #4.

[dimbear]

12. Counterexamples. The bane of mathematics.

Perhaps they should be outlawed. They've destroyed so much beautiful work.

[NinetySix]

5. In mathematical terms, perhaps...

The assertion:

"If 0/0 isn't equal to 25, then 0 is equal to 0"

is true in terms of statement logic, however. When set up as a hypothetical in the way you formulate it, "if p then q," a true antecedent entails a true consequent. Since zero cannot be used as a denominator, the statement "0/0 isn't equal to 25" is true, necessitating the truth of the consequent, "0 is equal to 0," according to the modus ponens form, "If p, then q; assert p, therefore q."

It is for this reason that the statement is likely considered to be both true and axiomatic, rather than as false or meaningless.

[Boojatta]

7. Kick

[Orrex]

8. Look. If you keep asking questions like this, I am SO not going to prom with you.

[Boojatta]

9. Kick

[bananas]

13. Related poll: "Have you studied model theory?"

After I saw this thread, I did a poll: Poll question: Have you studied model theory?
Only 3 people said they did (out of 11 total votes).