Is the API a 3rd party on or made by you?

If it is yours, consume the data sent by your form should be as simple as this on your API:

<?php
    $data = json_decode($_REQUEST['json']);

    echo $data[0]; // Should output 500
    echo $data[1]; // Should output hello world

If it is a 3rd party, probably they expect you to send the json in post body. To accomplish that, follow this post: How to post JSON to PHP with curl.

In main.php, use this code:

<?php
    $params = json_decode(file_get_contents("php://input"));
    echo $params->key_1;
?>

When you decode a JSON string, you convert it to an object of stdClass.

If you want to decode JSON and convert it to an array, use code below:

<?php
    $params = json_decode(file_get_contents("php://input"), true);
    echo $params['key_1'];
?>

What the heck are you trying? It looks as if you are trying to put key-value-pairs to the JavaScript-Array with arr.push('1', 'Brussels|25'); expecting an array containing "Brussels|25" under the key "1" - but watch out, you are creating this array: ["1", "Brussels|25", "2", "Antwerp|40"].

If you want to send json, send json-data:

var arr= [{  
    "city" : "Brussels",  
    "age" : 25  
},{  
    "city" : "Antwerp",  
    "age" : 40  
}];  

then your ajax call:

$.ajax({
   type: "POST",
   url: "jsondecode.php",
   data: {arr: JSON.stringify(arr)},
   success: function(data){
        console.log("success:",data);},
   failure: function(errMsg) {
        console.error("error:",errMsg);
   }
});

So you don't need to explode the data server-sided.

The server-sided script:

<?php
$data = json_decode($_POST["arr"]);
// will echo the JSON.stringified - string:
echo $_POST["arr"];
// will echo the json_decode'd object
var_dump($data);
//traversing the whole object and accessing properties:
foreach($data as $cityObject){
    echo "City: " . $cityObject->city . ", Age: " . $cityObject->age . "<br/>";
}
?>

Hope, this helps you now.

@edit: By the way, use console.log() or console.error() instead of alert. Alert will cause scripts to pause until you click on ok and you cannot see objects in an alert.

@2nd edit: the script is now tested, I removed unnecessary code and added the server-sided code

You can't send the JSON array directly, but you can prepare it in a way that allows it to be sent to your API. You turn your JSON into a string and encode it so it will be properly formatted for a GET.

// your JSON data
var data = '{"name":"Homer Simpson","status":"Suspended","disciplinary-dates":["2018-01-2","2018-02-14","2018-03-17"]}' ;

// take your JSON, turn it into a string and then encode it 
var urlParams = encodeURIComponent(JSON.stringify(data)) ;

// your URL adding your JSON data as the value
var url = 'http://example.com/?params=' + urlParams ;

On the PHP side you will decode

  // note: this code does not have any error checking - you should add it      
  // get your params
  $params = $_GET['params'] ;

  // decode the string
  $decodedParams = urldecode($params) ;

  // turn your string into an array
  $wasJSONAsArray = json_decode($decodedParams, true) ;

  // turn your string into a std object
  $wasJSONAsStdObj = json_decode($decodedParams) ;

If your whole POST body contains the JSON, you can get it using thid piece of code:

$json = file_get_contents('php://input');
$decoded = json_decode($json);

If you're not using a javascript library (jQuery, prototype.js, etc) that will do this for you, you can always use the example code from json.org

I think JSON.stringify() might be useful.

or, you can use json_decode() in php file.