Ideally, if you change out elements of a set with another iteration of the same element then you still have the same set. That's impractical with physical materials, but it's "close enough." More below.
Theseus' Ship is an interesting excercise in the notions of identity, but for one, the neurons we die with are essentially the ones we're born with -- if we're speaking of the self that is our mind. Even if they weren't, it seems reasonable that if a transition is gradual it doesn't matter, since the mind is an emergent property -- emergent properties don't particularly "care" if the proverbial elements of the set that compose it are switched out with the same.
You still have the same problem at hand. You're assuming the elements you swapped inside the set are the 'same'.
What if I went back, took all the discarded pieces of the original ship, and decided to reassemble another ship exactly like the original. It would now appear that both the 'restored ship' and the 'reassembled ship' to be equally qualified as the 'original'. Yet the two resulting ships are clearly not the same ship.
As a simplified example, consider a chess program -- a simple emergent thing with many analogues to a mind. Start switching out silicon chips in the computer and it doesn't really change what the program is, since part of what it means to be emergent is that it's based on the structure rather than the components -- so changing the components doesn't change what the emergent part is.
I don't know what you are trying to accomplish here. There are many roads to Rome, but that doesn't mean all roads are the same. I can make a computer program that obeys the same syntax of it's language, while having a different semantic block and having the same output. That does not mean both programs are the same.
I disagree -- introspection is immediate and direct. There's no time between starting and finishing the cogito. Perhaps it takes time to elucidate what the cogito is (such as, say, Descartes writing a book!); but for the being, it's instantaneous.
Even if it is, once you realize it, the next moment you become a different entity.
But this ignores that we have powerful justifications for the reality of non-dream states; and equally strong justification to believe the dream-states aren't real in the same sense as the waking world. It's rational to accept the waking world as real and the dreaming world as illusory. We don't require absolute sensory justification of the waking world to make that determination.
I already replied to this two posts ago. I will just post what I typed there.
Whatever we conceive in this state is a
subjective matter, Even if we have strong justification for it to be real. In my dreams, I can astral travel amongst the stars and do all sorts of impossibilities with strong justification of it being real. In my dreams, I am unable to discern if I am dreaming or not.
What makes you think atheists lack belief in ontological necessity? Your definition of atheism is flat out incorrect -- as are your assumptions of atheists' ontologies. Just because ontologically necessary things exist doesn't mean that they're "God" in any meaningful sense of that term.
The Earth is a necessary entity for us, but not for the Sun. The Sun is a necessary entity for Earth, but not for other celestial entities. Just because something can be a necessary-entity doesn't automatically make it a God-entity.
I started another post on this topic:
http://www.religiousforums.com/forum/religious-debates/125972-defining-god.html#post2738551
