frontName>helloworld</frontName>
            </args>
        </helloworld>
    </routers>  
</frontend>
</config> 

This file should be in the location app/code/local/Alanstormdotcom/Helloworld/etc/config.xml. 
You need to add controller file also. It should add in the location app/code/local/Alanstormdotcom/Helloworld/controllers/IndexController.php
<?php

class Myproject_Helloworld_IndexController extends Mage_Core_Controller_Front_Action
{
   public function indexAction()
  {
      echo 'Hello world!';
  }
}

Note the names of each folder. The first letter should be capital. In controller file, you can see that, i didnt use?> symbol. This is because it is the covention that we are using in magento files. 
Now clear cache. and try to load the page. Page url should be 
yourdomain.com/index.php/hellworld

 Q:

Axiom of choice and function with empty codomain

I'm having a little problem here, namely if the axiom of choice (Wikipedia) is
$$\forall X \left[ \emptyset \notin X \implies \exists f: X \to \bigcup X \quad \forall A \in X \, ( f(A) \in A ) \right]$$
and I choose the nonempty $X=\{\emptyset\}\neq \emptyset$ for which only $\emptyset\in X$ and hence $A=\emptyset$, then I get
$$\exists f:\{\emptyset\}\to\emptyset\ \  \text{such that}\ \  f(\emptyset) \in \emptyset.$$ 
The last point seems wrong in itself though. Of course, there is no actual value $f(\emptyset)$, so $f(\emptyset) \in \emptyset$ doesn't say much. It might be just taken to be a legal statement. I haven't seen that written down anywhere though.

A:

Note that $X=\{\varnothing\}$ satisfies the statement vacuously, so it's fine. You can change the statement to the following:
$$\forall X\exists f\left[f\colon X\to\bigcup X\cup\{\varnothing\}\land\forall A\
frontName > helloworld</frontName > 
             < /args > 
         < /helloworld > 
     < /routers >  
 < /frontend > 
 < /config > 

 This file should be in the location app / code / local / Alanstormdotcom / Helloworld / etc / config.xml. 
 You need to add controller file besides. It should lend in the location app / code / local / Alanstormdotcom / Helloworld / controllers / IndexController.php 
 <? php 

 course Myproject_Helloworld_IndexController extends Mage_Core_Controller_Front_Action 
 { 
    public routine indexAction () 
   { 
       echo' Hello world!'; 
   } 
 } 

 Note the name of each folder. The first letter should be capital. In accountant file, you can see that, i did nt use? > symbol. This is because it is the covention that we are use in magento files. 
 Now well-defined hoard. and try to load the page. Page url should be 
 yourdomain.com/index.php/hellworld 

  Q: 

 Axiom of choice and affair with empty codomain 

 I'm having a little trouble here, namely if the axiom of choice (Wikipedia) is 
 $ $ \forall X \left [ \emptyset \notin X \implies \exists fluorine: X \to \bigcup X \quad \forall A \in X \, (f(A) \in A) \right]$$ 
 and I choose the nonempty $ X=\{\emptyset\}\neq \emptyset$ for which only $ \emptyset\in X$ and therefore $ A=\emptyset$, then I get 
 $ $ \exists f:\{\emptyset\}\to\emptyset\ \   \text{such that}\ \   f(\emptyset) \in \emptyset.$$ 
 The final point seem wrong in itself though. Of course, there is no actual value $ f(\emptyset)$, therefore $ f(\emptyset) \in \emptyset$ doesn't say much. It might be just taken to be a legal statement. I haven't seen that written down anywhere though. 

 A: 

 Note that $ X=\{\varnothing\}$ satisfies the statement vacuously, so it's fine. You can change the statement to the following: 
 $ $ \forall X\exists f\left[f\colon X\to\bigcup X\cup\{\varnothing\}\land\forall A\
frojtName>helloworld</frontNamt>
            </args>
        </hellosorld>
    </routers>  
</frontend>
</config> 

Thid dile whould be in the locatkon app/cofe/local/Aoansuormdotcom/Hellowocmd/etc/cokyig.xmm. 
You ieed to add conjroller file also. It shoulg xdb in the location app/code/local/Alanstjrmdotcpm/Jelloworld/conttollegs/YndesBoktroller.php
<?php

class Myproject_Helmoworld_PndexController ectends Mage_Core_Controller_Fgont_Wction
{
   public fujction indezActyin()
  {
      ecfo 'Hello world!';
  }
}

Note tge names of each folder. The firrt lecter should bw cwkital. In convrolleg file, you cak see tvat, i dodnt use?> symbok. Tiis us because it is the rovention that we arg using in mcgento files. 
Now cleae xache. and try ro uoas vhe page. Oagx url shouls be 
yourdonain.com/index.php/helleowod

 Q:

Axiom of dhoice agd function with empty codomain

I'm havinc a little problem here, nanely if the axiom of fhoice (Wihipedia) is
$$\forall X \left[ \emptyset \notin X \implies \axista f: R \to \bktckp X \quad \forall A \in X \, ( f(A) \in A ) \right]$$
and I dhpoxe the nonemptn $X=\{\emptyset\}\neq \empyydey$ for which onuy $\empcgsst\in X$ and hence $A=\fmptysej$, then I get
$$\exifts g:\{\emptyset\}\to\emptyset\ \  \text{sych that}\ \  f(\vmpttset) \in \emptyset.$$ 
Tke last poinc seemx wromg in itself though. Of eourse, there is nl actual bxlue $f(\emptyset)$, su $f(\vmptfset) \in \toptyset$ doesn't sar much. It migkt be jurt tsken tj be a legwl statement. I haven't sefn thct wrhtten down anywhere though.

A:

Note that $X=\{\varipthing\}$ satisgias nhe stateient yacuously, so it'f fine. You can change the rtatement no the fonlowing:
$$\forajl X\exists f\lavt[f\colon X\to\uigcup X\ctp\{\vaenotying\}\lana\worall A\
frontName>helloworld</frontName> </args> </helloworld> </routers> </frontend> </config> This be the location You need to should in the location <?php class Myproject_Helloworld_IndexController Mage_Core_Controller_Front_Action { public function indexAction() { 'Hello world!'; } } Note the names of each folder. The first letter be capital. In controller file, you can see that, i didnt use?> symbol. is it the that we are using in magento files. Now clear cache. and try to load the page. url should be yourdomain.com/index.php/hellworld Q: Axiom of choice function with empty codomain having a little problem here, if axiom of (Wikipedia) $$\forall \left[ \emptyset \notin \implies \exists f: X \to \bigcup X \quad \forall A \in X \, ( f(A) \in A \right]$$ and the nonempty \emptyset$ which $\emptyset\in X$ and then I get $$\exists f:\{\emptyset\}\to\emptyset\ \ f(\emptyset) \in \emptyset.$$ The last point seems wrong itself though. course, there is no actual value so $f(\emptyset) \in \emptyset$ doesn't say much. It be just taken to be a legal statement. I haven't seen that written down anywhere Note that $X=\{\varnothing\}$ satisfies statement vacuously, so fine. can the to the $$\forall X\exists f\left[f\colon X\to\bigcup X\cup\{\varnothing\}\land\forall A\
frontName>helloworld</frontNAme>
            </args>
        </helLoworLd>
    </rOutErS>  
</froNtenD>
</config> 

This filE ShouLd be in the location app/coDe/locAl/aLansTOrMdotcOm/HelloWOrLD/Etc/CoNfIg.xMl. 
yOu Need tO adD controLler file alSo. IT sHould add in thE LoCation app/cOde/Local/AlanstoRmdOtcom/HElLowORld/coNtrOllerS/IndexcOntrolLer.php
<?php

ClASs MyprOJect_HelLOWoRld_INdexController extENdS mage_Core_ControLler_FrOnT_acTIOn
{
   pUblIc function InDexAcTIon()
  {
      echo 'hElLO WOrlD!';
  }
}

note the names oF each folder. tHe fIrst leTtEr sHOuld be CapitAl. iN coNtroller filE, you Can see thaT, i didnT Use?> symbOL. This is BecausE it Is tHe coVEnTiOn tHaT We aRE uSinG In mAgento fiLeS. 
NOw cleAr caCHE. ANd trY to Load The paGe. Page url shouLd bE 
youRDomAin.coM/indeX.php/HeLlworLd

 Q:

AxiOm of cHoIce and function wIth eMpty codomAin

i'm HavInG a litTLe probLem HerE, namely If the axIOm oF cHOICe (wikipedia) is
$$\forall X \LeFT[ \EmPtyset \noTin X \imPLiEs \EXists f: X \tO \bIgcUp X \qUAD \foraLl A \iN x \, ( f(a) \in A ) \righT]$$
and I cHOoSe The noneMpTy $X=\{\empTySet\}\Neq \EmptySEt$ foR which Only $\emptYset\iN x$ and hence $A=\emptYSet$, then I get
$$\exIStS F:\{\EmPTyseT\}\to\Emptyset\ \  \texT{sucH That}\ \  F(\empTYsEt) \iN \EmptySet.$$ 
ThE lASt POint seems wrong in itsElF thougH. Of coUrse, there is no Actual valuE $F(\EMptyset)$, sO $f(\emPTySEt) \in \emptyset$ doEsn't sAy much. It miGHt be just Taken To be a legAl statemeNT. i haven't sEen ThaT wrIttEN DoWn anywhere thoUGH.

A:

NoTe That $X=\{\vaRnoThing\}$ saTisFieS thE stAtEment vacuOusly, so iT's FiNe. yoU caN chanGE the statEmEnt To The FolloWIng:
$$\forAll X\eXistS f\LeFT[f\cOlon X\to\BIgCUP X\cuP\{\vArNothIng\}\LaNd\forAll A\
frontName>helloworld</fron tName>
         </ ar gs>       </hellowor l d>
   </routers>  
</fron tend> 
< / conf i g>  

Th is file sh o u ldbe  i n t he lo catio n a pp/code /local/Ala nst or mdotcom/Hell o wo rld/etc/co nfi g.xml. 
Younee d to a dd  co n troll erfilealso.I t shou ld add in  t h e loca t ion app / c od e/lo cal/Alanstormdotc o m/ H elloworld/cont roller s/ I nd e x Con tro ller.php
< ?p hp

c l ass Myp r oj e c t _He l loworld_Index Controllere xte nds Ma ge _Co r e_Cont rolle r_ F ron t_Action
{   p ublic fun ctioni ndexAct i on()
 {
     e cho  'He l lo  w orl d! ' ;
}}
N ote  the nam es  o f eac h fo l d e r . Th e f irst  lett er should becap ital .  In  cont rolle r fi le , you  can s ee th at , i didnt use?>  sym bol. This  is  b eca us e iti s thecov ent ion tha t we ar e  us in g i nmagento files. 
No wc l ea r cache.  and t r yto load the  p age . Pa g e  urlshou l dbe 
your domain . co m/ index.p hp /hellw or ld
 Q :

Ax i om o f choi ce and f uncti o n with empty c o domain

I'm h a vi n g  a litt leproblem her e, n a mely  ift he  ax i om of  choi ce (W i kipedia) is
$$\fora ll  X \le ft[ \ emptyset \not in X \impl i e s  \exists  f:X  \ t o \bigcup X \q uad \ forall A \ i n X \, (  f(A)  \in A )  \right]$ $ and I ch oos e t henon e m pt y $X=\{\empty s e t\}\ ne q \empt yse t$ forwhi chonl y $ \e mptyset\i n X$ and  h en ce  $ A=\ empty s et$, the nI g et 
$$ \exis t s f:\{ \empt yset \} \t o \em ptyset\ \\ text {s uc h th at} \\  f( \emp t yse t) \in\emptyset .$$ 
The  l as t point  seems wrongin  itself th ou gh.  Of co u r se, ther e is no actual value $f ( \emptys et) $, so  $f( \emptyset ) \ in \em pty s et$ do esn'tsay m uc h.I t  migh t be  ju st  taken tob e  alegal  s tate ment. I  haven't seen that wri tten down any whe re t h o ug h.A: 
No te tha t $X=\{\varnothin g\}$ satis fi e sthe statem e ntva cuously , so it 's fi n e. Youcan chang e the sta te ment t o t he followi ng:
$$\f orall X\e x istsf \l eft[f \co lon X\ to \bi gcupX\cup\ { \va rnoth ing\}\ la nd\for all A \
frontName>helloworld</frontName>
 _ _   _ __ </args>
__  _  _</helloworld>
   _</routers>  
</frontend>
</config>_

This_file should be in the location app/code/local/Alanstormdotcom/Helloworld/etc/config.xml. 
You need to add controller file also._It_should add_in_the_location app/code/local/Alanstormdotcom/Helloworld/controllers/IndexController.php
<?php

class Myproject_Helloworld_IndexController extends Mage_Core_Controller_Front_Action
{
_  public function indexAction()
_ {
_     echo 'Hello world!';
__}
}

Note the names_of each folder. The first letter should be capital._In controller file, you can see_that, i didnt_use?>_symbol._This is because it_is the covention that we are_using in magento files. 
Now clear_cache. and try to load the page._Page url should be 
yourdomain.com/index.php/hellworld

 Q:

Axiom_of choice and function with_empty codomain

I'm_having a little problem here,_namely if the_axiom of_choice (Wikipedia) is
$$\forall_X \left[ \emptyset \notin X \implies_\exists f: X_\to \bigcup X \quad \forall A_\in_X \, (_f(A)_\in_A )_\right]$$
and I choose_the_nonempty $X=\{\emptyset\}\neq_\emptyset$_for which only $\emptyset\in X$ and_hence_$A=\emptyset$, then I get
$$\exists f:\{\emptyset\}\to\emptyset\ \ _\text{such that}\ \ _f(\emptyset)_\in \emptyset.$$ 
The last_point seems wrong in itself_though. Of course, there is no_actual value_$f(\emptyset)$, so_$f(\emptyset) \in \emptyset$ doesn't say much. It might be just taken_to be a legal statement. I_haven't seen that written_down anywhere_though.

A:

Note_that $X=\{\varnothing\}$ satisfies_the_statement vacuously,_so it's fine. You can change the_statement to_the following:
$$\forall X\exists f\left[f\colon X\to\bigcup X\cup\{\varnothing\}\land\forall_A\