Things like this are not quite “particular integers” though in the informal sense (19 is centrally a “particular integer”, BB(1000) isn’t). They are more like integer-definitions whose meaning (as particular integers) can’t be characterized in some ways.
Similarly, the decision that an agent will eventually make is not a “particular decision” from the point of view of that agent, but a decision-definition whose meaning as a particular decision that agent can’t yet divine, because it’s not yet decided. Some external oracle might know more about the meaning of the decision-definition and predict what the decision will be before the agent makes the decision. And a stronger theory might “know” (in a given sense) which particular integer an integer-definition formulated in a weaker theory designates, or at least have some better characterization of it available.
Things like this are not quite “particular integers” though in the informal sense (19 is centrally a “particular integer”, BB(1000) isn’t). They are more like integer-definitions whose meaning (as particular integers) can’t be characterized in some ways.
Similarly, the decision that an agent will eventually make is not a “particular decision” from the point of view of that agent, but a decision-definition whose meaning as a particular decision that agent can’t yet divine, because it’s not yet decided. Some external oracle might know more about the meaning of the decision-definition and predict what the decision will be before the agent makes the decision. And a stronger theory might “know” (in a given sense) which particular integer an integer-definition formulated in a weaker theory designates, or at least have some better characterization of it available.