By Michael Potter
"Michael Potter offers a accomplished new philosophical creation to set idea. somebody wishing to paintings at the logical foundations of arithmetic needs to comprehend set thought, which lies at its middle. What makes the e-book distinct is that it interweaves a cautious presentation of the technical fabric with a penetrating philosophical critique. Potter doesn't simply expound the speculation dogmatically yet at each level discusses intimately the explanations that may be provided for believing it to be actual. Set conception and its Philosophy is a key textual content for philosophy, mathematical good judgment, and computing device science."
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"Michael Potter offers a accomplished new philosophical advent to set conception. someone wishing to paintings at the logical foundations of arithmetic needs to comprehend set concept, which lies at its center. What makes the e-book detailed is that it interweaves a cautious presentation of the technical fabric with a penetrating philosophical critique.
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Extra info for Set Theory and Its Philosophy: A Critical Introduction
X Φ(x ), which is read ‘the x such that Φ(x )’, to refer to the unique object which is Φ if there is one, and to nothing otherwise. Expressions of this form are called deﬁnite descriptions. More generally, expressions of the sort that denote objects are called terms. If Φ(x , x1 , . . , xn ) is a formula depending on the variables x , x1 , . . x Φ(x , x1 , . . , xn ) is a term depending on x1 , . . , xn . Proper names are also terms, but they do not depend on any variables. We shall use lower-case Greek letters such as σ, τ, etc.
During the 40 years since we have formulated with sufﬁcient precision the axioms of [set theory] and drawn their consequences in the most varied branches of mathematics, we have never come across a contradiction, and we are entitled to hope that one will never be produced. (Bourbaki 1954, p. 8) Of course, it is impossible to deny that the century which has elapsed without a contradiction being found is psychologically inﬂuential in engendering conﬁdence in the system. But this can scarcely be regarded as amounting to very much.
In the meantime we shall focus on another possible motivation based on the idea that there is a fundamental relation of presupposition, priority or, as we shall usually say, dependence between collections. The conception of the theory it gives rise to is now known as the iterative conception. In contrast with the limitation of size conception, it took a long time to emerge. It is not mentioned directly in Russell’s (1906a) discussion of possible solutions to the paradoxes, and the writings of the 1920s supply no more than glimmerings of it.
Set Theory and Its Philosophy: A Critical Introduction by Michael Potter