When we say that we possess a table, then we have a tangible idea of what this means: I own an x, such that x is a table, and that specific table present to me now is the referent for the name “table” where x = table. But what about something non-physical, such as love? Or numbers? Or freedom? Or boredom? And so on.

What an abstract number is can be thought of as a set X, such that the elements of X are all of the things equinumerous to one another. Thus, the number 2 is the set X such that all the elements of X are all the pairs in existence and anything that is not equinumerous to these pairs is not an element of the set. We might expand this further and say that X contains all of the possible things that are equinumerous pairs: not only is my holding up of my index finger and middle finger an element of X, but so is any logical possibility of my doing so.

Could we then apply this to something like love?

Let’s say that love is a set L, such that the elements of L are all of things experiencing love. We might even add the same extension as before: L also contains all of the logically possible experiences of love (i.e. the possible love I could feel for every combination of things which could possibly exist). Thus, my love for my mom, for my dad, for a good nap, for a delicious meal, and so on, are all elements of L, along with all possible love I (and all possible love-experiencers) could feel for all other possible things.

I think most people would agree that love can come in degrees (as well as types, such as platonic love, erotic love, the love of non-sentient things, and so on, but I will assume all these types of love describe elements of L). For instance, I might love pepperoni pizza more than mushroom pizza, even though I love both; I might love a friend I’ve known for 20 years more than one I’ve known for 2 years; and so on. And so, we could then generate a series out of the elements of L based on the love-magnitude relationship the elements have with objects inside and outside the group. For instance, lets say we have

x_{m} ∈ L

y_{n} = y_{v} + y_{u} ≔ y_{v} ∈ L ∧ y_{u}∉ L

Rx_{m}y_{n} = M_{mn}

Where R is the love relationship between x_{m} and y_{n} and M_{mn} is some magnitude of the love that x_{m} feels for y_{n}. And so, if Rx_{1}y_{1} is some love relationship between elements x_{1} and y_{1} with magnitude M_{11} and Rx_{1}y_{2} is some love relationship between x_{1} and y_{2} with magnitude M_{12} where M_{11} < M_{12} then we could order Rx_{m}y_{n} based on the relative magnitude M_{mn} such that every R combination of x_{m} with y_{n} is ordered in increasing M_{mn} in the series S. This would satisfy transitivity (M_{11} < M_{12} ∧ M_{12} < M_{13} ⊢ M_{11} < M_{13}), asymmetry (M_{11} < M_{12} ≠ M_{12} < M_{11}), and connectivity (M_{11} < M_{12} ⊢ M_{11} precedes M_{12} in S). And thus we get the full well-ordered series S composed of the elements (M_{11} , M_{12} , M_{13} , … , M_{1n} , M_{11} , M_{21} , M_{31} , … , M_{m1}) into a series of increasing magnitude.

Presumably M_{mn} would be some finite value, due to their being finite elements in L with finite magnitudes, and so it would not have infinite cardinality. But what is the nature of this finite value? Is there some finite maximum value that M_{mn} can take, or could it conceivably be infinite? The ontological argument for the existence of God would say that M_{mn} belongs to Anselm’s “the thing than which no greater can be thought.” Would it also be the case, then that

M_{mn} > S_{M}

Where

S_{M} = Σ^{m-1}_{i=1} Σ^{n-1}_{j=1} (Rx_{i}y_{j}) = the sum of all the magnitudes in S except for M_{mn}

But what reason is there for leaving M_{mn} out of S_{M}? L is the set of all love relationships, and so if S_{M} is the serialization of all the elements of L, then S_{M} must also include M_{mn}. But this would mean that

L > M_{mn}

And if we said that M_{mn} is defined as Anselm’s God, then this is contradictory, since L would be greater than “the thing than which no greater can be thought” and so would the pairing of M_{mn} and L (just in case one argued that M_{mn} and L are separate sets). And so, in order for God’s love to be a member of L, we would have to define L as a set containing a subset *greater* than the set into which the subset is contained:

M_{mn} ⊆ L ∧ M_{mn} > L

Which does not satisfy the property of reflexivity, since the finite cardinality of the subset M_{mn} is *larger* than the finite cardinality of the set L. This means 1) that there is no 1-to-1 mapping of points in the subset M_{mn} to points in the set L and 2) the axiom of regularity is violated if we say that M_{mn} = L, which would make:

(M_{mn} ⊆ L ∧ M_{mn} = L) → (M_{mn} ⊆ L ∧ M_{mn} = M_{mn}) → (M_{mn} ∈ M_{mn})

Thus, this is one more reason why the ontological argument doesn’t work.