The tree property for non-weakly compact $\kappa$ In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$.
One of which was that $\kappa$ is inaccessible and has the tree property (that is if any tree of cardinality $\kappa$ for which every level is of cardinality $<\kappa$ then it has a branch (i.e. a maximal chain) of cardinality $\kappa$). 
I can understand the property itself and what it means. However, since $\aleph_1$ or $\aleph_\omega$ are clearly not weakly-compact cardinals, there should be a tree which contradicts this property.
How do you build this sort of tree?
 A: What you are looking for is the concept of Aronszajn tree. You can read about constructions of Aronszajn trees in any graduate level set theory text, and meanwhile, the Wikipedia page lists a summary of the basic facts:

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*Kőnig's lemma states that $\aleph_0$-Aronszajn trees do not exist.


*The existence of Aronszajn trees ($=\aleph_1$-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of König's lemma does not hold for uncountable trees.


*The existence of $\aleph_2$-Aronszajn trees is undecidable (assuming a certain large cardinal axiom): more precisely, the continuum hypothesis implies the existence of an $\aleph_2$-Aronszajn tree, and Mitchell and Silver showed that it is consistent (relative to the existence of a weakly compact cardinal) that no $\aleph_2$-Aronszajn trees exist.


*Jensen proved that $V=L$ implies that there is a κ-Aronszajn tree (in fact a κ-Suslin tree) for every infinite successor cardinal κ.


*Cummings & Foreman (1998) showed (using a large cardinal axiom) that it is consistent that no $\aleph_n$-Aronszajn trees exist for any finite n other than 1.


*If κ is weakly compact then no κ-Aronszajn trees exist. Conversely if κ is inaccessible and no κ-Aronszajn trees exist then κ is weakly compact.
Finally, let me point out a small inaccuracy in your question. The equivalence is that $\kappa$ is weakly compact iff it is inaccessible and has the tree property. It is not correct to drop the inaccessibility part, as you did, since it is consistent that $\aleph_2$ has the tree property.
A: Some results supplementing Joel's answer:

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*Shelah proved (around 1995) that if $\lambda$ has cofinality $\omega$ and is the supremum of strongly compact cardinals, then $\lambda^+$ has the tree property. See


Menachem Magidor, and Saharon Shelah. The tree property at successors of singular cardinals, Archive for Math Logic, 35 (5-6), (1996), 385-404. MR1420265 (97j:03093).


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*Neeman proved that, assuming the existence of $\omega$ supercompact cardinals, we can force a model where the tree property holds at all the $\aleph_n$ ($2\le n<\omega$) and at $\aleph_{\omega+1}$.  See


Itay Neeman. The tree property up to $\aleph_{\omega+1}$, preprint.

Neeman's result improves previous results, both in terms of the cardinals with the tree property, and in consistency strength: Magidor and Shelah had obtained the tree property at $\aleph_{\omega+1}$ from a huge cardinal with $\omega$ supercompact cardinals above. As mentioned in Joel's answer, Cummings and Foreman had obtained the tree property for the $\aleph_n$ ($2\le n<\omega$), also from $\omega$ supercompact cardinals. At the moment, Neeman's is the best current result in terms of intervals of regular cardinals with the tree property. At least in $\mathsf{ZFC}$.

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*Arthur Apter proved (around 2009) that the following is consistent, relative to a proper class of supercompact cardinals: $\mathsf{ZF} + \mathsf{DC} +$ Every successor cardinal is regular and has the tree property, while every limit cardinal is singular. See these slides, and


Arthur W. Apter. A remark on the tree property in a choiceless context, Arch. Math. Logic, 50 (5-6), (2011), 585–590. MR2805298 (2012d:03115).

The conclusion of Apter's result implies determinacy in $L(\mathbb R)$, and more.

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*The upper bound in consistency strength for successive cardinals with the tree property is a supercompact cardinal with a weakly compact cardinal above it. Around 1983, Abraham forced, from these assumptions, that $2^{\aleph_0}=\aleph_2$, and both $\aleph_2$ and $\aleph_3$ have the tree property. All results on successive cardinals with the tree property build on Abraham's argument. See


Uri Abraham. Aronszajn trees on $\aleph_2$ and $\aleph_3$, Ann. Pure Appl. Logic, 24 (3), (1983), 213–230. MR0717829 (85d:03100).


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*The best known lower bound is due to Foreman, Magidor, and Schindler. They show that if all $\aleph_n$ ($2\le n<\omega$) have the tree property, and $\aleph_\omega$ is strong limit, then $\mathsf{PD}$ holds. See


Matthew Foreman, Menachem Magidor, and Ralf Schindler. The consistency strength of successive cardinals with the tree property, J. Symbolic Logic, 66 (4),(2001), 1837–1847. MR1877026 (2003m:03083).

This result is frustrating in the sense that we expect two successive cardinals with the tree property should give us much more in consistency strength than this, beyond $\mathsf{AD}^{L(\mathbb R)}$, and likely beyond the current reach of descriptive inner model theory. Still, this would be frustratingly short of the best current upper bounds, which experts expect are much closer to the truth.
