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For first order logic without equality, what are the exact axioms we give to define the relation of equality ?

I can't find the exact axioms even in Wikipedia... Any reference to any article or a list of the exact axioms would do ...

  • See Equality and its axioms and see also this post. – Mauro ALLEGRANZA Jan 26 '24 at 11:05
  • If we want to formulate e.g. first-order arithmetic using as underlying logic predicate calculus without equality, we have to start with at least $a =b \to (a=c \to b =c)$ that allows us to prove the key properties of equality: $a=a$, $a = b \to b = a$ and $a = b \to ( b = c \to a = c )$ and then we have to prove the substitution property for the three dunction symbols: $s, +, \times$. See e.g. Kleene, Introduction to Metamathematics (1952), Ch.VIII Formal Number Theory. – Mauro ALLEGRANZA Jan 26 '24 at 11:12
  • @MauroALLEGRANZA from the post you linked in the comment, it's the 2nd and 3rd axiom I have difficulty understanding, since the symmetric and transitive property for equality is defined with x=y as a premise itself... how to actually prove if a given x is actually equal to a given y.... I admit this sounds naive, but I am really having a hard time understanding it, maybe I am making a very silly mistake in understanding it... –  Jan 26 '24 at 12:24
  • @MauroALLEGRANZA given the above doubt, I am facing a dilemma where to me it looks like equality isn't even definable or axiomatizable in first order logic... how to actually show or under what axioms x = y , unless we actually know what x and y are we shouldn't be able to give a general axiom or proof of x = y.... –  Jan 26 '24 at 12:31
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    "it looks like equality isn't even definable or axiomatizable in first order logic" Why? the axioms listed in Wiki's entry are the axioms forequality. – Mauro ALLEGRANZA Jan 26 '24 at 12:36

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