can any one give me a first-order sentence that says there exist exactly three elements. Please explain it to me.
Thanks
How about this (assuming you mean that there exist exactly three distinct elements):
$\displaystyle (\exists x)(\exists y)(\exists z)\left[\neg(x=y)\land\neg(x=z)\land\neg(y=z)\land(\forall t)((t=x)\lor(t=y)\lor(t=z))\right].$
This says that there are three elements that are mutually unequal, and if you consider any element at all, it must be equal to one of those three elements. Does that do the trick?
Thank you very much about that. I found this in a book but without example about the first order sentence that says there are three elements.
So , as I understood , I just write a sentence that has three elements which are distinct. is that correct
Thanks
Well, if you're trying to say that there are exactly three elements, then you have to have a way to say that there aren't any more than three. That's what I'm doing with the t in my expression above. So, the first three AND'ed clauses ensure that you have at least three distinct elements. The t business ensures that everything is one of the three elements you've already constructed, thus guaranteeing that there aren't any more than three elements.
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Fernando Revilla