Example of A Provable Identity
Since the axioms seem to list all the basic facts about the operations in question it is not immediately obvious that there should be anything one can state using only the three operations that is not provably true. However, proving seemingly innocuous statements can require long proofs using only the above eleven axioms. Consider the following proof that (x + 1)2 = x2 + 2 · x + 1:
- (x + 1)2
- = (x + 1)1 + 1
- = (x + 1)1 · (x + 1)1 by 9.
- = (x + 1) · (x + 1) by 8.
- = (x + 1) · x + (x + 1) · 1 by 6.
- = x · (x + 1) + x + 1 by 4. and 3.
- = x · x + x · 1 + x · 1 + 1 by 6. and 3.
- = x1 · x1 + x · (1 + 1) + 1 by 8. and 6.
- = x1 + 1 + x · 2 + 1 by 9.
- = x2 + 2 · x + 1 by 4.
Here brackets are omitted when axiom 2. tells us that there is no confusion about grouping.
The length of proofs is not an issue; proofs of similar identities to that above for things like (x + y)100 would take a lot of lines, but would really involve little more than the above proof.
Read more about this topic: Tarski's High School Algebra Problem
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