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Isn't the only formal analytic definition of God, that of Cantor's Absolute Infinity?

What is the state-of-the-art of this approach?

Are there other definitions?

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    The "prime mover" comes to mind - the first cause in every complete chain of cause and effect.
    – aschepler
    Commented Mar 25 at 20:08
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    Cantor's "bigger than any other conceivable or inconceivable quantity, either finite or transfinite" is hardly formal or analytic. It is a variation on Anselm's "that, than which nothing greater can be conceived", which is also neither and just as vacuous. The "state of the art" is that "formally defining" God is nonsensical. Formal definitions need undefinable primitive notions, and what would serve as that for "God"? The best one can do is implicit 'definition' through axioms like Gödel's for the ontological argument.
    – Conifold
    Commented Mar 25 at 20:35
  • Godel defined x as being God, if x has the quality that every positive quality is true for x Commented Mar 26 at 2:24
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    The absolute in the Grundlagen (1883) is linked to God: "finiteness is part of the concept of number and that the true infinite or Absolute, which is in God, permits no determination whatsoever." But there is no "formal" definition of God; see footnote: "All things finite or infinite are definite and, God excepted, can be determined by the intellect." Commented Mar 26 at 7:39
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    See Michael Hallett, Cantorian Set Theory and Limitation of Size (Clarendon Press, 1984) for the complete discussion. Commented Mar 26 at 7:56

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Horsten[16] (see also Barton[22]) compares the use of the concept of absolute infinity in theology to its use in theories of proper classes. (Horsten also happens to have written the current SEP article on the philosophy of mathematics in general.) With respect to Cantor's own theology, consider:

When I deal with the 'transfinite', the absolute infinite is not meant, which as actus purus and ens simplicissimum neither can be increased nor decreased and only exists in Deo or rather as Deus optimus maximus." ... What lies above all finite and transfinite is not a 'Genus'; it is the only completely individual unit within which all is contained, which comprises all, the 'absolute', incomprehensible to human intellect, therefore not subject to mathematics, the 'ens simplicissimum', the 'actus purissimus', which by many is called 'God'. [emphasis added]

So as the New Advent Encyclopedia entry on divine attributes reads:

In this respect, the relation of the attributes to the Divine nature might be illustrated by the various reflections of one and the same object from a concave, a convex, and a plane mirror. Nevertheless, to systematize the idea of God, and to draw out the rich content of the knowledge resulting from the proofs of God's existence, some primary attribute may be chosen as representing one aspect of the Divine perfection from which the others may be rigorously deduced. Then arises a logical scheme in which the derivative attributes, or perfections stand towards one another in a relation somewhat similar to that of the essence and the various properties and qualities in a material substance. In this arrangement the primary perfection is termed the metaphysical essence, the others are called attributes. The essence, too, may be regarded as that characteristic which, above all others, distinguishes the Deity from everything else. Upon the question, which attribute is to be considered primary, opinions differ. Many eminent theologians favour the conception of pure actuality (Actus Purus), from which simplicity and infinity are directly deduced. Most modern authors fix on aseity (Aseitas; a = "from" se = "himself"), or self-existence; for the reason that, while all other existences are derived from, and depend on, God, He possesses in Himself, absolutely and independently, the entire reason of His uncaused infinite Being. In this, the most profound and comprehensive distinction between the Divinity and everything else, all other distinctions are implicitly expressed. Whether, and in what way, the distinctions between the attributes and the metaphysical essence, and among the attributes themselves have an ontological basis in the Divine nature itself was subject which divided Nominalists and Realists, Thomists and Scotists, in the age of Scholasticism (cf. Vacant, Dict. de théol. cathol., I, 2230-34). [emphasis added]

Gödel's ontological argument involves a "definition" of God in terms of what Kant denoted the ens realissimum:

This conception of a sum-total of reality is the conception of a thing in itself, regarded as completely determined; and the conception of an ens realissimum is the conception of an individual being, inasmuch as it is determined by that predicate of all possible contradictory predicates, which indicates and belongs to being. ... The possibility of things must therefore be regarded as derived- except that of the thing which contains in itself all reality, which must be considered to be primitive and original. For all negations- and they are the only predicates by means of which all other things can be distinguished from the ens realissimum—are mere limitations of a greater and a higher—nay, the highest reality; and they consequently presuppose this reality, and are, as regards their content, derived from it. The manifold nature of things is only an infinitely various mode of limiting the conception of the highest reality, which is their common substratum; just as all figures are possible only as different modes of limiting infinite space. The object of the ideal of reason—an object existing only in reason itself—is also termed the primal being (ens originarium); as having no existence superior to him, the supreme being (ens summum); and as being the condition of all other beings, which rank under it, the being of all beings (ens entium).

It is not entirely unproblematic to proclaim the divine nature to be absolutely infinite. The SEP entry on omnipotence reads:

Transfinite cardinals may be used to quantify an amount of energy or force, e.g., ℵ0 joules, or ℵ0 newtons. Or they may be be used to state the cardinal number of a totality of objects, e.g., ℵ2 states of affairs. Thus, one [first] way in which ‘power’ might be interpreted is as a power (or range of powers) to produce an energy or force quantified by a transfinite cardinal. One such option is that ‘infinite power’ means the power to produce a specific transfinite quantity of joules or newtons, e.g., ℵ0. The other option is that ‘infinite power’ means the power to produce any transfinite quantity of joules or newtons, i.e., ℵ0, ℵ1, ℵ2, and so on ad infinitum. A third, more radical option, is that ‘infinite power’ means the power to produce more joules or newtons than can be quantified by any transfinite cardinal number. Such energy or force would appear to qualify as energy or force than which none greater is possible. ... The standard view is that there is no greatest transfinite cardinal, and that there are proper classes containing more objects than can be quantified by any transfinite cardinal. Moreover, there could not be an agent who has greater power than an omnipotent agent. Thus, with regards to the two interpretations offered, there is some reason to conclude that in each case the third option is best. ... Even so, the notions of more energy or force than can be quantified by any transfinite cardinal, and of more states of affairs than can be quantified by any transfinite cardinal, are mind-boggling. Moreover, arguably, that much energy or force, or the power to bring about that many states of affairs, is impossible. For this reason, with respect to each of the interpretations of ‘infinite power’ on offer, there are doubts about the viability of the corresponding third option. Thus, with respect to each of these alternate interpretations of ‘infinite power’, it is an open question which of the three corresponding options is best. Until greater clarity is attained about this matter, a definition of omnipotence as infinite power is problematic.

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  1. It is well-known that neither a biggest cardinal number nor a biggest ordinal number exist. I assume that also Cantor, who invented transfinite set theory, was aware of this result from his theory.

    I do not understand from the references what Cantor meant by “Absolute infinity”. Apparently he does not speak as a mathematician.

  2. Under the assumption that defining “God” as the “Absolute infinity” is a definition at all, that is not an analytic definition. Because one cannot find absolute infinity as a property of the term "God" by analyzing the meaning of the term.

    Instead I consider the proposal to equate the concept of “God” with the concept of “Absolute Infinity” an example of explaining obscurum per obscurius (to explain the obscure by the more obscure).

  3. For an overview of the different definitions of “God” which have been proposed during the history of philosophy see the Stanford online encyclopedy God, Chap. 2.2.

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  • Good answer. Well deserved +1 Commented Mar 25 at 21:04
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    For "Absolute" in Cantor, see Grundlagen (1883) Commented Mar 26 at 7:40
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    It is not well-known that there is no largest cardinal, see Gitman, et. al. [11]: "The counterexample ZFC- models we produce can be arranged to satisfy various natural strengthenings of ZFC-, to include such statements as ... 'I am Hθ+,' meaning the assertion that θ is the largest cardinal and every set has hereditary size at most θ." Granted, that is in a nonstandard system of set theory, but so there is nothing absolutely intrinsic to set theory that rules out a largest cardinal as such. Commented Mar 26 at 11:22
  • @KristianBerry You are correct, but I think we should be more fair to the poster. If someone says it is well known that the number 1 is finite- it is a bit pedantic to say, well consider this nonstandard formulation of arithmetic. I think we can give the poster the benefit of the doubt of speaking within the framework if standard set theory, even if they don't explicitly say so. Commented Mar 29 at 23:28
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Classical Theism offers multiple definitions Classical Theism operates off definitions and absolutes. God as creator of everything. God as the perfection of all attributes. God as the only necessary object/entity. God as an Omni-being (Omniscient, Omnipotent, Omni-benevolent). Note these are not equivalent definitions or criteria, and Abrahamic Gods may not actually satisfy any/all of these criteria.

Problems with Classical Theism Definitions For each classical definition, there are varying refutations. Necessary being? There are multiple God conceptions, they can't all be the only necessary being, and as they are multiple, WHAT KIND of God, is clearly a contingent question. So none are necessary. Omni-being? Omni-beings get into all sorts of logic problems. So rather than absolute "omni" these problems can be resolved by just treating Omni as "very very much" rather than an infinite term. But very very much beings still fail the optimization tests like Problem of Evil, problem of contradictory, imperfect, or false communications, etc.

"Perfections" is clearly a judgement call, as to what is or isn't a "perfection" of an attribute. I have read arguments in multiple directions on multiple "perfections". So this is not a credible or useful attribute, much less a clear and obvious absolute. Creator of everything, including time, causation, reasoning, logic, etc is -- problematic, as that is a causation and state sequence logic, which is part of what has yet to be created .... Also, God does not get one out of the Munchausen Trilemma.

Pragmatic/Empirical Approach There is also another approach to "defining" God, and that is the loose empirical/pragmatic approach, of offering a partial working definition that can be used for most purposes, and then refined as needed. Using this approach, one can realize that the Gods of most religions generally do not satisfy the criteria of classical theism, but DO satisfy the following categories:

  • Non-material/spiritual agent
  • With large power to affect the physical universe, much more so than humans
  • And significant knowledge about how the universe works, much more so than humans

Naturalizing God claims, making them pragmatic and limited, and modifiable based on test cases, rather than these inflexibly theory-based absolutes, extends naturalism into the spiritual realm. It also leaves the spiritual in the same state as the physical -- each discovery of "what is fundamental" in physics, leaves the next unanswered question -- "and why is THAT the case" -- which we currently do not know for the Standard Model of QM. Likewise God as a hypothesis leaves open "how did we get THIS God" for further investigation

Takeaway One can use both sets of criteria to do test cases. The classical theism tests are generally failed, mostly because they are absolutes, and our world does not seem to satisfy absolutist theory. The more limited pragmatic list -- is also testable, but is not clearly refuted (although it also struggles with the problems of non-optimization, so any God conception that one applies them to has to be modified in ways to explain them).

Directions for Further Research The pragmatic method of treating God as a testable and modifiable hypothesis is used in two movements I am aware of: Gnostic mysticism, and Process Theology. I have a few examples of each I can offer, as jumping off points for further research.

For gnostic mysticism, I offer an example of Ben Swett, and his empirical/gnostic method of treating spirituality as an exploration: bswett.com/spirit.html

For Process theology, I offer two examples: Thomas Oord, and his limited God: amazon.com/God-Cant-Believe-after-Tragedy/dp/1948609126 and Nancy Abrams with A God Who Could be Real: publishersweekly.com/9780807073407.

Anyone who wants to find more explicit theological references in either movement can take these as a jumping off point.

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  • The classical theism tests are generally failed - In your opinion, what are the top 3 books making this case?
    – Mark
    Commented Mar 25 at 20:49
  • Thanks. How about paper references?
    – Mark
    Commented Mar 26 at 14:01
  • none of this is formal
    – ac15
    Commented Mar 26 at 14:49
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    My point is that [are there more than one God claims?] is irrelevant to [Are multiple versions of God claims logically possible?]. If the answer to the second is "yes" it wouldn't matter if there was only one God claim. If the answer to the second is "no, there is exactly one" it doesn't matter if there are a trillion God claims: at most one of them is correct.
    – g s
    Commented Mar 27 at 3:16
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    @ScottRowe -- only if it is accompanied by an assertion of "necessity". If one accepts that no God conception is "necessary", contingent God conceptions can be treated like all other hypotheses.
    – Dcleve
    Commented Apr 3 at 14:44
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Let's first settle on one of the best examples of a formal definition many of us are familiar with: the first few pages of The Elements by Euclid. My hope is that the synopsis I am about to describe briefly brings most of us, not just the expert and well versed, into a similar enough mental state to communicate effectively on this question.

The Elements start with a long list of definitions for point, line, plane.., (a). Note that some of those definitions are somewhat mysterious like the one for a "point:" that which is of no-dimensions, or alternatively, that which has no magnitude. Then come the famed five axioms like all right angles are equal (b). It then lists briefly some shared (c) concepts, like the transitivity of equivalence of quantities and the fact that parts are not greater than their sum. And only then, finally, we get into the never ending propositions or theorems, like the fact that two of the angles in an isoceles triangle are equal, or, the sum of angles of a triangle is equal to the sum of two right angles, so on, some of which may seem "obviously true," yet Euclid formalized their proof by basing it on more fundamental notions we've mentioned earlier.

Now, please also note that it took more than two millennia to discern what was special about the last axiom and how we could eliminate it and get to other kinds of geometries, like spherical or hyperbolic planes.

Next, consider reality -- the grand complexity we all experience seemingly encompassing the ideal, reasoned world of geometry, and math and computation, in general, but also the empirical world of physics, chemistry, biology, behavioral psychology and AI, or alternatively, electrons, water, trees, bees, birds and bonobos, chatgpt and cyborgs -- the reality for which we are yet to even imagine any full formalism despite the best attempts of the few TOEs, theories of everything, posited by thinkers and theoretician over the brief history of humanity.

So, given all of that which we are yet to formalize, and given the challenge of even formalizing geometry as I attempted to describe briefly above, it does seem to be nonsensical to attempt to define Reality or God in any formal way.

"God is Reality" may be a tautology, but, I think, it is helpful to remember when we desire, nay, crave, on and off, to get a grasp on either side of that tautology. I would suggest some introspection: who or what is attempting to grasp who or what? Can the knife cut itself? Can the eye see itself? Can the I know Oneself? Eventually and inevitably.

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