Differences between real numbers in $ACA_0 + \lnot Con(PA)$ and standard real numbers? Let $M = (\mathbb N_M, 0_M, +_M, \times_M, <_M, D_M)$ be a model of $ACA_0 + \lnot Con(PA)$. We define $\mathbb R_M$ as Dedekind cuts on $D_M$.
What can we say about the differences between $\mathbb R_M$ and $\mathbb R$? We know for example that B-W fails in $\mathbb Q_M$. I also suspect it to fail for $\mathbb R_M$ if we represent sequences of real numbers by function formulas $\phi(n,x)$, where $n$ is a natural number and $x$ is a set representing a real number.
What other differences are there?
 A: You can find many examples about the topology of the reals in $ACA_0$ in Simpson's $\it Subsystems\ of\ Second$-$\it Order\ Arithmetic$. I don't think $Con(PA)$ itself has a usable topological interpretation, but since $ATR_0$, the next Big Five system above $ACA_0$, does prove $Con(PA)$, you could also look at the chapter on $ATR_0$; those theorems should be false in your system.
$ACA_0$ is strong enough to prove $\it most$ properties of the reals that you would find in an intro to real analysis. To use an example you mentioned already, the Bolzano-Weierstrass theorem is provable in $ACA_0$ (for the Dedekind cuts $\mathbb{R}_M$): For any sequence $\langle r_1,r_2,\ldots\rangle$ of cuts in $[0,1]$, you can zero in on an convergent subsequence by iteratively asking the arithmetical question, 'Are infinitely many $r_i$ in the sequence bigger than my chosen value $q$?'.
One well-known theorem that won't hold in your system are the Cantor-Bendixson Theorem (requires $\Pi^1_1$ comprehension); or the lesser-known, related Perfect Set Theorem for closed sets (requires $ATR_0$).
