Not only are the axioms to which we apply logic variable enough for someone in a closed system of belief to be warranted, on their own terms, in thinking that there is a God, but logic itself is variable enough to open the door to personally justified inferences to the existence of such a God. As Onora O'Neill put it (channeling Immanuel Kant), we are permitted to hope that it is possible for there to be a God, though we might not be obligated to claim to know that there actually is a God:
... any adequate account of what we may hope will have to incorporate some account of anything that we must hope. There might, however, be many distinct answers to the question “What may I hope?” that had in common only those aspects of hope that are required. It may, for example, be the case that various quite distinct hopes for human destiny incorporate a convincing account of what we must hope. ... Even if the abstract claims of deism and the tenets of traditional Christian faith provide two specific answers to the question “What may I hope?” Kant’s arguments may not show that either forms part of every answer to the narrower question “What must I hope?”
Perhaps there are conclusive public justifications for some axioms and inference rules, such that we might publicly disavow the admissibility of belief in God as such; but you note that you are asking about, and focusing on, more private cases. By analogy, one might adopt a set of idiosyncratic axioms in set theory, and/or a peculiar logic (maybe a system with 19 truth values, one connective (TONK), etc.), and go on to arrive at peculiar conclusions. Those inferences will be relatively justified to some extent; of course, if it came time to apply them to the judgments and in turn actions of other people, other considerations will come into play. But if it's just oneself and one's own opinions that are at stake, then yes, it does seem acceptable to hope for, or maybe even believe in, the possibility or actuality of God's existence.
Moreover, lest one overzealously claims to have come up with a perfect, consensus definition of divine attributes, let them read through the lack of consensus (LOC) regarding the following:
- LOC in definitions of "omniscience."
- LOC re: "omnipotence."
- LOC re: "perfect goodness."
- LOC re: "omnipresence."
- LOC re: "divine eternity."
- LOC re: concepts of God or "ultimates" in general.
Historically, it was not always the case that theologians worked primarily from pre-theoretical conglomerates of lower-order predicates, to define their God, but the appeal was more deeply made to formal predicates of a superlative character:
... the concept of “infinite being” has a privileged role in Scotus’s natural theology. As a first approximation, we can say that divine infinity is for Scotus what divine simplicity is for Aquinas. It’s the central divine-attribute generator [emphasis added]. But there are some important differences between the role of simplicity in Aquinas and the role of infinity in Scotus. The most important, I think, is that in Aquinas simplicity acts as an ontological spoilsport for theological semantics. Simplicity is in some sense the key thing about God, metaphysically speaking, but it seriously complicates our language about God. God is supposed to be a subsistent simple, but because our language is all derived from creatures, which are all either subsistent but complex or simple but non-subsistent, we don’t have any way to apply our language straightforwardly to God. The divine nature systematically resists being captured in language.
For Scotus, though, infinity is not only what’s ontologically central about God; it’s the key component of our best available concept of God and a guarantor of the success of theological language. That is, our best ontology, far from fighting with our theological semantics, both supports and is supported by our theological semantics.
We can even see this with respect to Aquinas' model of a triune deity, where the model is built from a theory of relations (polyadic predicates) modulo an essentialism (and essence/existence distinction) inherited from Aristotle. (C.f. the difference between (A) a philosophy of mathematics that takes at least the natural numbers for granted and (B) one which seeks to ground 0 and ℕ in something else (e.g. Frege's peculiar introduction of the empty class as the class of items with contradictory properties, or Cantor's transmutation of ℕ into the ordinal ω and the cardinal ℵ0).)
Addendum: Gödel's God
I should like to add that Kurt Gödel's ontological argument was read off the notion of an ens realissimum (c.f. Olivier Esser's positive set theory), which I will account for by quoting Kant:
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. It is, therefore, a transcendental ideal which forms the basis of the complete determination of everything that exists, and is the highest material condition of its possibility—a condition on which must rest the cogitation of all objects with respect to their content. Nay, more, this ideal is the only proper ideal of which the human mind is capable; because in this case alone a general conception of a thing is completely determined by and through itself, and cognized as the representation of an individuum.
Kant goes on to explain that this primal unity is a cognitive structure, something carved into the architectonic of reason in seeking to simplify and condense its elementary premises as much as possible. So it is not, "just like that," the mark of an actual God. Nevertheless, even per Gödel's reasoning later, we can see that the definition of an ens realissimum is intuitively noncontradictory by itself, either because it is defined in terms of quintessential consistency in abstracto/from the outside, or because the definition is too generalized and simplistic to license enough substantial inferences to infer a substantial contradiction from. Modulo the incompleteness theorems, we would say, "The theory of an ens realissimum does not allow us to derive enough of arithmetic for the incompleteness problem to confront the theory; the theory is so weak that its consistency would be provable, and then is too simple to be proven contradictory." Rather than just coherent definitions of God being epistemically possible, then, it seems as if we know of a coherent definition of God that is already actual, if perhaps trivial (in the negative limit).