Where this gets really confusing and mind bending is that the tic-tac-toe game exists both in the Shoggoth's story and in the real world. I really am playing tic-tac-toe, but against a fictional guy. I'm playing against no one. Claude is fake, and the Shoggoth doesn't know we're playing.
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Isn't that how people work as well? Telling a story about a fictional character until they believe that's who they are. I say and do the things I say and do to fit the narrative I have created about who I am.
This isn't to say that human identities are not real...
In a sense, that created narrative / persona is a reflection of our own internal Shoggoth, just as it is for an AI.