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Can a collection be any single object in itself, for example Collection A is 'one collection' or set A is 'one set' or even one 'mathematical object' if viewed as a whole? For example a set containing 3 objects can be one 'collection' or one 'set' but in the end it is comprised of three objects that take their own continuous volume.

Confused
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    The point of view of set theory is YES: we have properties of the collection (example: number of its elements) that are not properties of the elements. – Mauro ALLEGRANZA May 25 '22 at 15:56
  • To "think" a collection as a single "unity" we have to consider some property that the objects have: to be a satellite of Earth (a collection of one object) vs to be a satellite of Jupiter (80?) – Mauro ALLEGRANZA May 25 '22 at 16:09
  • Yes. All things, among other ontological properties, are collections, from multiple perspectives (a house can be a set or building materials or a set of rooms). A rock is a collection of particles. A particle is a collection of atoms. An atom is a collection of quarks. A quark is a set of concepts. A concept is a set of abstract ideas. An idea is a set of thoughts. A thought is a set of judgements. A judgement is a set of considerations. This never ends: all mental objects are tautologically sustained on others, like language in the dictionary is circularly defined by the same set of words. – RodolfoAP May 25 '22 at 17:12
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    @MauroALLEGRANZA I assume you know that mathematical sets need not be characterized by a property. A standard example is a choice set on the equivalence classes of the reals mod the rationals. That's a set, by the axiom of choice; but there is no property that determines whether a particular real number is in it. – user4894 May 25 '22 at 18:04
  • @user1007028: Too many questions, and a mess of concepts; it is better to address problems by posting clear and precise questions. Anyway: yes; no; yes; void; yes. – RodolfoAP May 27 '22 at 03:02
  • @RodolfoAP Will try to be clearer next time – Confused May 27 '22 at 09:18
  • Complex query which has been around a long time. See: Merology: https://plato.stanford.edu/entries/mereology/ https://en.wikipedia.org/wiki/Mereology https://plato.stanford.edu/entries/location-mereology/ – gonzo May 28 '22 at 23:34
  • @user1007028 "a set containing 3 objects" A set is a collection of things, not a box "containing" things. We say correctly "a set of three objects", not "a set containing three objects". Talk of "set containing members" is bad language imposed by the absurd mathematical notion of empty set implying that a set is a kind of box. – Speakpigeon May 29 '22 at 10:16
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    @Speakpigeon Do you allow the mathematical existence of nonempty sets, such as the set N of natural numbers? And do you accept the axiom schema of specification? Then you must necessarily accept the mathematical existence of the set ∅ = {x ∈ N : x ≠ x}. What say you? If you don't believe in the empty set, which premise do you reject? The set of natural numbers? Specification? Logic? Help me to understand your claim. https://en.wikipedia.org/wiki/Axiom_schema_of_specification – user4894 Nov 30 '22 at 20:54
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    @user4894 "Do you allow the mathematical existence of nonempty sets, such as the set N of natural numbers?" I don't know what mathematical existence. There is no problem with ℕ. 2. "you must necessarily accept the mathematical existence of the set ∅ = {x ∈ N : x ≠ x}" No. The expression x ≠ x is false so the definition of ∅ is nonsensical. Russell's paradox is evidence that Russell and mathematicians don't know how logic works or are happy to adopt false solutions if it is expedient to do so on the moment. I have no issue with defining subsets out of sets. – Speakpigeon Dec 01 '22 at 17:27
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    @Speakpigeon You need to understand the axiom schema of specification. Clearly you do not. – user4894 Dec 01 '22 at 19:18
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    @Speakpigeon ps -- Do you understand that x ≠ x is a predicate with one parameter? It's true for some things (in this case, no things) and false for some things (in this case, all things). But it's still a one-parameter predicate. You plug in a value for x and the predicate takes a truth value. Can you see that? Can you articulate your argument to the contrary? Are you saying x ≠ x is not a one-parameter predicate? It's no different in principle than x = x, which I assume you recognize as a predicate that's true for all assignments of x. x ≠ x is a predicate that's false for all assignments – user4894 Dec 01 '22 at 21:05
  • @user4894 "It's no different in principle than x = x, which I assume you recognize as a predicate that's true for all assignments of x." It is different. "x ≠ x is a predicate that's false for all assignments" See? – Speakpigeon Dec 02 '22 at 14:53
  • @user4894 "You need to understand the axiom schema of specification. Clearly you do not." You need to understand logic. Clearly you do not. – Speakpigeon Dec 02 '22 at 14:55
  • Do sets exist? A nuanced old answer showing the continuum of possibilities – Rushi Jul 29 '23 at 05:13
  • @Speakpigeon, nice, really, really, reeaallly nice! – Agent Smith Jul 29 '23 at 05:21
  • @user4894, ditto! – Agent Smith Jul 29 '23 at 05:21
  • Mathematically speaking ... 12 eggs = 1 dozen, so on and so forth I suppose. – Agent Smith Jul 29 '23 at 05:23

3 Answers3

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The question whether a collection can be a single ‚thing‘ recalls the relation between a whole and its parts:

Is the whole more than the sum of its parts?

If the answer is yes for a given whole, than the whole should be considered a new entity. A typical example is a system, or more specific an organism.

Jo Wehler
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  • a "whole" may not exhaust the individuality of "collections", but this seems the case. It comes up in Buddhist studies, especially old studies, whether a collection has to be a whole (given that wholes do not exist). I believe the standard answer is that all real collections - ones not constructed by the mind - are wholes, but I never understood why. –  Aug 03 '22 at 12:26
  • In support of this answer: "Set-theoretic mereology," Hamkins and Kikuchi [2016]. – Kristian Berry Jul 28 '23 at 23:03
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For example a set containing 3 objects can be one 'collection' or one 'set' but in the end it is comprised of three objects that take their own continuous volume

This is not necessarily true. A quarter (single object) is equivalent to 25 pennies but a quarter does not occupy the same volume as 25 pennies. Furthermore, most vending machines will accept a quarter but not 25 pennies so in this application, the quarter is a single indivisible object even though it is equivalent to 25 pennies.

  • I guess in that case we separate the 'physical' collection of 25 pennies and the 'physical' entity of one quarter which is not a collection of other coins (it is but one coin) I would say they have the same magnitude of abstract (money) that is measured in a number of how many 'units' of currency we count. – Confused Jul 07 '22 at 09:39
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Mathematically speaking ... 12 eggs = 1 dozen, so on and so forth I suppose.

Agent Smith
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