By having no beliefs about gods one way or the other
If they have no beliefs about gods, it cannot be said that there is no entity they consider not to be a god. For to consider whether some entity is x or has properties y is to believe that the entity is x or has those properties y. If an atheist has no beliefs about god, it cannot be said that there is anything they consider
not to be a god.
... e.g. by having the mindset of a newborn baby.
Who, because your definition of "god" relies entirely on subjective consideration, are incapable of believing there exists anything that they don't consider to be a god.
Where did your arbitrary "there must be at least one thing an individual believes in" criterion come from?
Negations with existential and universal quantification.
Consider a universe of all possible things. A theist believes there exists an entity they consider to be (a) "god". We can say this 2 ways:
Let Gx = "is a god".
A theist is defined by the their belief in either of the following (which are logically equivalent) :
~∀(x)~Gx (i.e., it is not the case that for any x whatsoever, x isn't god)
or
∃(x) Gx (i.e., there exists an (i.e., at least one) x such that x is god)
The negations of either of these statements define all atheists according to you because they are the complement to the set of theists:
∀(x)Gx (i.e., for any x whatsoever, x isn't god)
or
~∃(x) Gx (i.e., there exists no x such that x is god)
The only way for someone to be an atheist and not deny that "there exists an x such that x is god" is to believe that for any x
1) x is not a god
2) it is not true that x is not a god
This is not possible unless you want to extend beyond classical logic, set theory, etc. (it won't help you out here, but as is you have a contradiction that is only resolved if all atheists deny there is any x they consider to be a god and therefore that deny any gods exist)
Another way to put that is to say that atheists are not the complement set to the set "theists", for if they were, then it must be true of all atheists that they believe there exists no x such that x is god AND (equivalently) that for any x whatsoever, x isn't god. That's a denial that any gods exist, and would be the strict, logical complement to your set of theists: denial that any gods exist.
If you make unjustified logical leaps, sure. They let you do lots of things.
You defined the set. If you have the problem with my logic, use of set theory, or are otherwise inclined to believe me inaccurate, then point out the logical flaws. For example, must a theist believe that these (logically equivalent statements) are true:
~∀(x)~Gx (i.e., it is not the case that for any x whatsoever, x isn't god)
or
∃(x) Gx (i.e., there exists an (i.e., at least one) x such that x is god)
If not, then your set of theist allows those who believe "there exists no god" to be members.
The logical equivalent of "there exists a god" is "it is not the case that there exists no gods". The negation of these is "there exists no god" and "it is the case that no gods exists".
If you can give an alternative logical, consistent definition of your set that makes my statements illogical, by all means do. However, defining atheists as the complement to the set theists doesn't allow for much range. It is a logical complement, which means a logical negation. If a member of the set "theists" must believe "(a) god exists", then the complement to the set of theists is
by definition those who believe the
negation of that statement: "no god exists".
Someone who has no beliefs about god does not belong to the complement of the set of those who believe in at least one god any more than a rock does. Put more formally, they are not in the domain of the membership function that determines membership in the set "theists". That function decides whether any individual is a theist if and only if they believe in something they consider gods. Sets require well-defined (bijective or one-to-one) mappings that evaluate membership as either in or out. The set "theists" who believe in something they consider god maps the beliefs of every individual regarding any entity they believe in as either
1) believing in an entity considered god (theist)
2) not believing in any entity
considered god
Someone with no beliefs can't be said to not believe in any entity they consider to be (a) god, because if there were
any entity they considered not to be god then they would have beliefs.