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I'm looking for someone that can explain this lecture slide to me in a more clear language. I have an exam in a few days and i just cannot wrap my head around why this means that there is only one x s.t. A(x).
In my understanding, you take ∀y for A(y) and then cross-reference ∀x and ∀y to find the value of x and y that both satisfy A(x). However, i don't clearly see what this value of y is, because to me it seems completely arbitrary based on what you choose for y and therefore ∀x∀y((A(x)∧A(y))→ x = y ) does not exclude the possibility of multiple x values that can satisfy A(x) as there can be multiple instances where x=y.
If anyone can give me an example or tips on how to visualize this it would really helpfull!

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    NO; assume that there are x1 and x2 such that both satisfy A. This means that A(x1) ∧ A(x2) is true and so from the formula above you will derive - using Modus Ponens - that x1=x2. Jan 21 at 16:38
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    "At most" because the formula is true also when there are none. But if there are, they all "coincide" (see argument above). Jan 21 at 16:52
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    Check with the natural numbers (starting from 0) and with the property "less than 2". You will see that the formula is not true. Jan 21 at 16:55
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    See Uniqueness quantification Jan 21 at 17:18
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    Exactly...... but not "random" sets, but elements whatever of the same "domain" or universe of discourse. Jan 21 at 17:54

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[...] does not exclude the possibility of multiple x values that can satisfy A(x) as there can be multiple instances where x=y.

There is only one value which is equal to x, and that's x itself. If x = y, then x and y are the same thing, because that's the definition of the equals symbol. Therefore, you can't have multiple "instances" where x = y, because the only value that y can take on in order to satisfy this equation is x.

Sometimes, I have seen people use the equals symbol to mean "not actually the same thing, but close enough that you can't tell them apart." However, in philosophy and (usually) mathematics, we consider indiscernible objects to be identical - that is, if you can't tell them apart, then they are the same thing. On the other hand, in software engineering and (to some extent) computer science, there is a meaningful distinction between object identity and object equivalence, and the equals sign is sometimes co-opted to refer to the latter rather than the former. It must be understood that this is an abuse of notation, and the equals symbol used in this context does not have the same meaning as the symbol when used by mathematicians and philosophers. You will also sometimes see mathematicians using the equals sign to represent objects which are isomorphic rather than actually identical, which is usually not too much of a problem but could lead to confusion in some cases.

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  • It is all quite problematic. Atoms (of the same kind) are identical to each other. But you can have two or more of them, so they are different items. Yet they are indistinguishable and interchangeable. In mathematics, we think that there is one and only one number "1". Everyone using it uses the same number.
    – gnasher729
    Jan 25 at 23:16
  • @gnasher729: Different atoms can be distinguished from each other by virtue of being in different places or at different quantum states (e.g. spin, charge, etc.). Applying this reasoning to bosons is harder, however.
    – Kevin
    Jan 25 at 23:55

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