Logically, must past events be finite in number?

  • Clarifying the question

    A ceiling labeled "0" has a chain hanging from it. Each link in the chain is marked "1, 2, 3" and a standing man is on the bottom of the chain.

    Is it philosophically impossible for an an actual infinity (ℵ0) of events to occur? Does reason alone allow us to conclude the past is finite in duration?

    Actual infinity = def. A completed infinity. A collection of definite and discrete members whose number is greater than any natural number. (In set theory, completed sets like {0, 1, 2, 3…} and {0, 2, 4, 8…} are actually infinite.)1

    1. The actual infinity is contrasted with the potential infinity.

      William Lane Craig & James Sinclair: “Cantor called the potential infinite a “variable finite” and attached the sign ∞ (called a lemniscate) to it; this signified that it was an “improper infinite” (Cantor 1915, pp. 55–6). The actual infinite he pronounced the “true infinite” and assigned the symbol ℵ0 (aleph zero) to it. This represented the number of all the numbers in the series 1, 2, 3, . . . and was the first infinite or transfinite number, coming after all the finite numbers. According to Cantor, a collection or set is infinite when a part of it is equivalent to the whole (Cantor 1915, p. 108). Utilizing this notion of the actual infinite, Cantor was able to develop a whole system of transfinite arithmetic.” [The Blackwell Companion to Natural Theology (Blackwell, 2009), 104.]

  • Experts often say yes

    • Rudgar Vaas: “Because we are able to assign a symbol to represent ‘infinity’ and can manipulate such a symbol according to specified rules, one might assume that corresponding infinite entities (e.g. particles or universes) exist. But the actual (i.e. realized in contrast to potential or conceptual) physical (in contrast to the mathematical) infinite has been criticized vehemently being not constructible, implying contradictions etc. … If this would be correct it should also apply to an infinite past. [“Time before Time.” arxiv.org/pdf/physics/0408111 (2004), 9.]
    • George Ellis, Uli Kirchner, & William Stoeger: “Can there be an infinite set…? We suggest that, on the basis of well-known philosophical arguments, the answer is No. [“Multiverses and Physical Cosmology.” arXiv:astro-ph/0305292 v3 (2003), 14.]
“Yes, after all…

  • Infinities can't exist in the world

    A hotel extends to the clouds. It has no bottom.

    In general, an actual infinity cannot exist or be instantiated in the mind-independent world.

    This page analyzes two arguments…

    The inability for actual infinities to be instantiated suggests the past is finite because an infinite past would require the instantiation of an infinite number of past events.1 As confessed by one atheist specialist on the topic:

    Graham Oppy: “…an infinite number of events stretching back into the past would form an actually infinite set… I am happy to grant that an infinite temporal regress is an actual infinite.” [Arguing about Gods (Cambridge, 2006), 141-142.]

    No,

    So? Couldn't it simply be that…

    • Presentism is true. If it is, past events no longer exist and so there really is no presently instantiated actual infinity. (But see reply).1
    1. Even on presentism, there "have been" an actually infinite number of events. Some critics say this response implies the future is then also actually infinite (which might be an undesirable result), because by way of parallel reasoning there "will be" an actual infinity number of events. However, this assumption is misplacedl there will never be an actual infinity of future events. Rather, the series of events will go toward infinity as a limit. This is part an parcel with presentism where an actual and potential infinity are distinguished.

  • Infinities can't be formed by addition

    A collection formed by successive-addition cannot be infinite.1

    After all:

    • Any finite quantity plus another finite quantity is always a finite quantity.

    This is relevant because the temporal series of events in the past is a collection formed by successive addition.

    So? Plausibly…

    • Past events have always been infinite in number?
    • Past events are not successive-additions?2
    1. If an infinite number of hours can be “achieved,” then someone can have a infinitieth birthday. Absurdities and internal contradictions abound.

      • Ellis, Kirchner, & Stoeger: “The arguments against an infinite past time are strong – it’s simply not constructible in terms of events or instants of time, besides being conceptually indefinite” [G.F.R. Ellis, U. Kirchner, and W.R. Stoeger, “Multiverse and Physical Cosmology,” Monthly Notices of the Royal Astronomical Society, Vol. 347, Is. 3, (2004): 14.]

    2. For example:

      • Graham Oppy: “...if one supposes that time has the structure of the real numbers and if one also supposes that there are continuous processes in time, then one will deny that past events form a series, and one will also deny that the collection of past events fall under a relation of successive addition. … time is modelled by the real numbers in so many of our most successful scientific theories,…” [Arguing about Gods (Cambridge, 2009), 142-143 (Oppy said this also in 2006)]