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Although actual infinites can have an ideal existence, th... — Carmelics

Home/Natural Theology

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Although actual infinites can have an ideal existence, they cannot really exist.

Natural Theology

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Reasons For

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Reason for

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1.If actual infinites that neither increase nor decrease in the number of members they contain were to really exist, absurd consequences would result.
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2.In a library with an actually infinite number of books containing an infinite number of red and an infinite number of black books, the library would contain as many red books as the total books in its collection and as many red books as red and black books combined. But in reality the subset cannot be equivalent to the entire set.
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3.In a real library, removing a certain number of books reduces the overall collection. But if infinites are actual, a library with an infinite number of books would not be reduced in size at all by removal of a specific number of books, such as all the red books or those with even catalogue numbers.
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Reasons Against

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Reason against 1 of 2

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1.The 'absurdities' in Hilbert's Hotel and similar cases are merely counterintuitive results of transfinite arithmetic, not logical contradictions.
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2.Cantor's set theory rigorously demonstrates that a proper subset of an infinite set can have the same cardinality as the whole, which is a provable theorem, not an absurdity.
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3.An inference from 'violates finite intuitions' to 'cannot really exist' conflates psychological surprise with metaphysical impossibility.
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Reason against 2 of 2

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1.The argument illicitly assumes that operations governing finite collections, such as subtraction and part-whole relations, must govern infinite collections in the same way.
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2.Georg Cantor and contemporary set theorists like Quine and Maddy have shown that denying the part-whole axiom for infinite sets yields a consistent, well-developed mathematics that maps onto physical theories.
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3.If actual infinities are ruled out by Craig's reasoning, the continuous spacetime manifolds and infinite-dimensional Hilbert spaces employed in modern physics would also be metaphysically impossible, an implausible conclusion.
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Topics

Natural Theology

Related

An inference from 'violates finite intuitions' to 'cannot really exist' conflate...Cantor's set theory rigorously demonstrates that a proper subset of an infinite ...Georg Cantor and contemporary set theorists like Quine and Maddy have shown that...If actual infinites that neither increase nor decrease in the number of members ...

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If actual infinities are ruled out by Craig's reasoning, the continuous spacetim...In a library with an actually infinite number of books containing an infinite nu...In a real library, removing a certain number of books reduces the overall collec...The 'absurdities' in Hilbert's Hotel and similar cases are merely counterintuiti...The absurdities resulting from attempting to apply basic arithmetical operations...The argument illicitly assumes that operations governing finite collections, suc...

Similar

There can be no such thing as an actual infinite.85%If actual infinites that neither increase nor decrease in the number o...83%The absurdities resulting from attempting to apply basic arithmetical ...82%The existence of an infinite series in actuality is impossible81%

Source

AI-extracted3/3 agreementValid

SEP: cosmological-argument

Craig and Sinclair 2009; Craig and Smith 1993

View source passageHide passage

Since conclusion 8 follows validly, if premises 6 and 7 are true the argument is sound. In defense of premise 6, he defines an actual infinite as a determinate totality that occurs when a part of a system can be put into a one-to-one correspondence with the entire system (Craig and Sinclair 2009: 104). Craig argues that if actual infinites that neither increase nor decrease in the number of members they contain were to really exist, rather absurd consequences would result. For example, imagine a library with an actually infinite number of books. Suppose that the library also contains an infini...

Extraction notes

Validity: The premises accurately capture Craig's argument from the source passage—presenting the absurd consequences of actual infinites (library thought experiments involving subset equivalence and book removal) and concluding that actual infinites can have ideal but not real existence.

Confidence: High confidence. The argument is explicitly laid out in the text as Craig's defense of premise 6.

Details

Type: claim
Perspectives: 3 (1 for, 2 against)
Edits: 1 edit