Ising model phase transition

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Homework Statement


Can someone please explain to me why there can never be a phase transition in the 1D Ising model?


Homework Equations





The Attempt at a Solution


I have read the argument that if we start at T=0, all spins along the 1D chain are aligned (say up).
Then if we slightly increase T, we can get a so-called "grain boundary" with changes the energy by 2J, which remains finite in the thermodynamic limit (as the size of the chain becomes infinite). This is unlike the 2D case where the energy change associated with a grain boundary blows up as the system size becomes infinite. That all makes sense, but what does all this have to do with phase transitions?
 
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